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Physically, ")),Q("mjx-container",L,[(e(),a("svg",x,t[17]||(t[17]=[l('',1)]))),t[18]||(t[18]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",{mathvariant:"bold"},"x")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),t[21]||(t[21]=T(" encompasses the amplitudes of either point-like or collective oscillators (e.g., mechanical resonators, voltage oscillations in RLC circuits, an oscillating electrical dipole moment, or standing modes of an optical cavity)."))]),t[261]||(t[261]=Q("p",null,"As the simplest example, let us first solve the harmonic oscillator in frequency space. 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The system is not solvable in Fourier space anymore!"))]),t[262]||(t[262]=Q("h2",{id:"Harmonic-ansatz-and-harmonic-equations",tabindex:"-1"},[T("Harmonic ansatz & harmonic equations "),Q("a",{class:"header-anchor",href:"#Harmonic-ansatz-and-harmonic-equations","aria-label":'Permalink to "Harmonic ansatz & harmonic equations {#Harmonic-ansatz-and-harmonic-equations}"'},"​")],-1)),Q("p",null,[t[64]||(t[64]=T("Even though we need an infinity of Fourier components to describe our system exactly, some components are more important than others. 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Depending on the parameters, the number of real solutions is known to be between 1 and 3."))]),t[268]||(t[268]=Q("h3",{id:"Sidenote:-perturbative-approach",tabindex:"-1"},[T("Sidenote: perturbative approach "),Q("a",{class:"header-anchor",href:"#Sidenote:-perturbative-approach","aria-label":'Permalink to "Sidenote: perturbative approach {#Sidenote:-perturbative-approach}"'},"​")],-1)),Q("p",null,[t[196]||(t[196]=T("The steady states describe a response that may be recast as ")),Q("mjx-container",n3,[(e(),a("svg",o3,t[180]||(t[180]=[l('',1)]))),t[181]||(t[181]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("msub",null,[Q("mi",null,"x"),Q("mn",null,"0")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("msub",null,[Q("mi",null,"X"),Q("mn",null,"0")]),Q("mi",null,"cos"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"ω"),Q("mi",null,"d")]),Q("mi",null,"t"),Q("mo",null,"+"),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},")")])],-1))]),t[197]||(t[197]=T(", where ")),Q("mjx-container",d3,[(e(),a("svg",r3,t[182]||(t[182]=[l('',1)]))),t[183]||(t[183]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("msub",null,[Q("mi",null,"X"),Q("mn",null,"0")]),Q("mo",null,"="),Q("msqrt",null,[Q("msup",null,[Q("mi",null,"u"),Q("mn",null,"2")]),Q("mo",null,"+"),Q("msup",null,[Q("mi",null,"v"),Q("mn",null,"2")])])])],-1))]),t[198]||(t[198]=T(" and ")),Q("mjx-container",i3,[(e(),a("svg",h3,t[184]||(t[184]=[l('',1)]))),t[185]||(t[185]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"ϕ"),Q("mo",null,"="),Q("mo",null,"−"),Q("mtext",null,"atan"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"v"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mo",null,"/")]),Q("mi",null,"u"),Q("mo",{stretchy:"false"},")")])],-1))]),t[199]||(t[199]=T(". Frequency conversion from ")),Q("mjx-container",p3,[(e(),a("svg",g3,t[186]||(t[186]=[l('',1)]))),t[187]||(t[187]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("msub",null,[Q("mi",null,"ω"),Q("mi",null,"d")])])],-1))]),t[200]||(t[200]=T(" to ")),Q("mjx-container",H3,[(e(),a("svg",u3,t[188]||(t[188]=[l('',1)]))),t[189]||(t[189]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mn",null,"3"),Q("msub",null,[Q("mi",null,"ω"),Q("mi",null,"d")])])],-1))]),t[201]||(t[201]=T(" can be found by setting ")),Q("mjx-container",c3,[(e(),a("svg",w3,t[190]||(t[190]=[l('',1)]))),t[191]||(t[191]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"≡"),Q("msub",null,[Q("mi",null,"x"),Q("mn",null,"0")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"+"),Q("mi",null,"δ"),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),t[202]||(t[202]=T(" with ")),Q("mjx-container",L3,[(e(),a("svg",x3,t[192]||(t[192]=[l('',1)]))),t[193]||(t[193]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),Q("mi",null,"δ"),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),Q("mo",null,"≪"),Q("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),Q("msub",null,[Q("mi",null,"x"),Q("mn",null,"0")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])],-1))]),t[203]||(t[203]=T(" and expanding Eq. \\eqref{eq:duffing} to first-order in ")),Q("mjx-container",f3,[(e(),a("svg",V3,t[194]||(t[194]=[l('',1)]))),t[195]||(t[195]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"δ"),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),t[204]||(t[204]=T(". The resulting equation"))]),Q("mjx-container",y3,[(e(),a("svg",M3,t[205]||(t[205]=[l('',1)]))),t[206]||(t[206]=Q("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mi",null,"δ"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"x"),Q("mo",null,"¨")])]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"+"),Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("msubsup",null,[Q("mi",null,"ω"),Q("mn",null,"0"),Q("mn",null,"2")]),Q("mo",null,"+"),Q("mfrac",null,[Q("mrow",null,[Q("mn",null,"3"),Q("mi",null,"α"),Q("msubsup",null,[Q("mi",null,"X"),Q("mn",null,"0"),Q("mn",null,"2")])]),Q("mn",null,"4")]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")]),Q("mi",null,"δ"),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mo",null,"−"),Q("mfrac",null,[Q("mrow",null,[Q("mi",null,"α"),Q("msubsup",null,[Q("mi",null,"X"),Q("mn",null,"0"),Q("mn",null,"3")])]),Q("mn",null,"4")]),Q("mi",null,"cos"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mo",{stretchy:"false"},"("),Q("mn",null,"3"),Q("msub",null,[Q("mi",null,"ω"),Q("mi",null,"d")]),Q("mi",null,"t"),Q("mo",null,"+"),Q("mn",null,"3"),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},")"),Q("mstyle",{scriptlevel:"0"},[Q("mspace",{width:"0.167em"})]),Q("mo",null,",")])],-1))]),Q("p",null,[t[211]||(t[211]=T("describes a simple harmonic oscillator, which is exactly soluble. Correspondingly, a response of ")),Q("mjx-container",k3,[(e(),a("svg",Z3,t[207]||(t[207]=[l('',1)]))),t[208]||(t[208]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"δ"),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),t[212]||(t[212]=T(" at frequency ")),Q("mjx-container",b3,[(e(),a("svg",v3,t[209]||(t[209]=[l('',1)]))),t[210]||(t[210]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mn",null,"3"),Q("msub",null,[Q("mi",null,"ω"),Q("mi",null,"d")])])],-1))]),t[213]||(t[213]=T(" is observed. 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Physically, ")),Q("mjx-container",L,[(e(),a("svg",x,t[17]||(t[17]=[l('',1)]))),t[18]||(t[18]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",{mathvariant:"bold"},"x")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),t[21]||(t[21]=T(" encompasses the amplitudes of either point-like or collective oscillators (e.g., mechanical resonators, voltage oscillations in RLC circuits, an oscillating electrical dipole moment, or standing modes of an optical cavity)."))]),t[261]||(t[261]=Q("p",null,"As the simplest example, let us first solve the harmonic oscillator in frequency space. 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The system thus responds at the driving frequency only - the behaviour can be captured by a single harmonic. This illustrates the general point that ")),t[40]||(t[40]=Q("em",null,"linear systems are exactly solvable",-1)),t[41]||(t[41]=T(" by transforming to Fourier space, where the equations are diagonal."))]),Q("p",null,[t[44]||(t[44]=T("The situation becomes more complex if nonlinear terms are present, as these cause ")),t[45]||(t[45]=Q("em",null,"frequency conversion",-1)),t[46]||(t[46]=T(". 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The system is not solvable in Fourier space anymore!"))]),t[262]||(t[262]=Q("h2",{id:"Harmonic-ansatz-and-harmonic-equations",tabindex:"-1"},[T("Harmonic ansatz & harmonic equations "),Q("a",{class:"header-anchor",href:"#Harmonic-ansatz-and-harmonic-equations","aria-label":'Permalink to "Harmonic ansatz & harmonic equations {#Harmonic-ansatz-and-harmonic-equations}"'},"​")],-1)),Q("p",null,[t[64]||(t[64]=T("Even though we need an infinity of Fourier components to describe our system exactly, some components are more important than others. 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Notice that once ansatz \\eqref{eq:harmansatz} is used in Eq. \\eqref{eq:ode}, all terms become oscillatory - each prefactor of ")),Q("mjx-container",Q1,[(e(),a("svg",t1,t[74]||(t[74]=[l('',1)]))),t[75]||(t[75]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"cos"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"ω"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"i"),Q("mo",null,","),Q("mi",null,"j")])]),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),t[81]||(t[81]=T(" and ")),Q("mjx-container",T1,[(e(),a("svg",a1,t[76]||(t[76]=[l('',1)]))),t[77]||(t[77]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"sin"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"ω"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"i"),Q("mo",null,","),Q("mi",null,"j")])]),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),t[82]||(t[82]=T(" thus generates a separate equation. 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Depending on the parameters, the number of real solutions is known to be between 1 and 3."))]),t[268]||(t[268]=Q("h3",{id:"Sidenote:-perturbative-approach",tabindex:"-1"},[T("Sidenote: perturbative approach "),Q("a",{class:"header-anchor",href:"#Sidenote:-perturbative-approach","aria-label":'Permalink to "Sidenote: perturbative approach {#Sidenote:-perturbative-approach}"'},"​")],-1)),Q("p",null,[t[196]||(t[196]=T("The steady states describe a response that may be recast as ")),Q("mjx-container",n3,[(e(),a("svg",o3,t[180]||(t[180]=[l('',1)]))),t[181]||(t[181]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("msub",null,[Q("mi",null,"x"),Q("mn",null,"0")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("msub",null,[Q("mi",null,"X"),Q("mn",null,"0")]),Q("mi",null,"cos"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"ω"),Q("mi",null,"d")]),Q("mi",null,"t"),Q("mo",null,"+"),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},")")])],-1))]),t[197]||(t[197]=T(", where ")),Q("mjx-container",d3,[(e(),a("svg",r3,t[182]||(t[182]=[l('',1)]))),t[183]||(t[183]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("msub",null,[Q("mi",null,"X"),Q("mn",null,"0")]),Q("mo",null,"="),Q("msqrt",null,[Q("msup",null,[Q("mi",null,"u"),Q("mn",null,"2")]),Q("mo",null,"+"),Q("msup",null,[Q("mi",null,"v"),Q("mn",null,"2")])])])],-1))]),t[198]||(t[198]=T(" and ")),Q("mjx-container",i3,[(e(),a("svg",h3,t[184]||(t[184]=[l('',1)]))),t[185]||(t[185]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"ϕ"),Q("mo",null,"="),Q("mo",null,"−"),Q("mtext",null,"atan"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"v"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mo",null,"/")]),Q("mi",null,"u"),Q("mo",{stretchy:"false"},")")])],-1))]),t[199]||(t[199]=T(". Frequency conversion from ")),Q("mjx-container",p3,[(e(),a("svg",g3,t[186]||(t[186]=[l('',1)]))),t[187]||(t[187]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("msub",null,[Q("mi",null,"ω"),Q("mi",null,"d")])])],-1))]),t[200]||(t[200]=T(" to ")),Q("mjx-container",H3,[(e(),a("svg",u3,t[188]||(t[188]=[l('',1)]))),t[189]||(t[189]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mn",null,"3"),Q("msub",null,[Q("mi",null,"ω"),Q("mi",null,"d")])])],-1))]),t[201]||(t[201]=T(" can be found by setting ")),Q("mjx-container",c3,[(e(),a("svg",w3,t[190]||(t[190]=[l('',1)]))),t[191]||(t[191]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"≡"),Q("msub",null,[Q("mi",null,"x"),Q("mn",null,"0")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"+"),Q("mi",null,"δ"),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),t[202]||(t[202]=T(" with ")),Q("mjx-container",L3,[(e(),a("svg",x3,t[192]||(t[192]=[l('',1)]))),t[193]||(t[193]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),Q("mi",null,"δ"),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),Q("mo",null,"≪"),Q("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),Q("msub",null,[Q("mi",null,"x"),Q("mn",null,"0")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])],-1))]),t[203]||(t[203]=T(" and expanding Eq. \\eqref{eq:duffing} to first-order in ")),Q("mjx-container",f3,[(e(),a("svg",V3,t[194]||(t[194]=[l('',1)]))),t[195]||(t[195]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"δ"),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),t[204]||(t[204]=T(". The resulting equation"))]),Q("mjx-container",y3,[(e(),a("svg",M3,t[205]||(t[205]=[l('',1)]))),t[206]||(t[206]=Q("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mi",null,"δ"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"x"),Q("mo",null,"¨")])]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"+"),Q("mrow",{"data-mjx-texclass":"INNER"},[Q("mo",{"data-mjx-texclass":"OPEN"},"["),Q("msubsup",null,[Q("mi",null,"ω"),Q("mn",null,"0"),Q("mn",null,"2")]),Q("mo",null,"+"),Q("mfrac",null,[Q("mrow",null,[Q("mn",null,"3"),Q("mi",null,"α"),Q("msubsup",null,[Q("mi",null,"X"),Q("mn",null,"0"),Q("mn",null,"2")])]),Q("mn",null,"4")]),Q("mo",{"data-mjx-texclass":"CLOSE"},"]")]),Q("mi",null,"δ"),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mo",null,"−"),Q("mfrac",null,[Q("mrow",null,[Q("mi",null,"α"),Q("msubsup",null,[Q("mi",null,"X"),Q("mn",null,"0"),Q("mn",null,"3")])]),Q("mn",null,"4")]),Q("mi",null,"cos"),Q("mo",{"data-mjx-texclass":"NONE"},"⁡"),Q("mo",{stretchy:"false"},"("),Q("mn",null,"3"),Q("msub",null,[Q("mi",null,"ω"),Q("mi",null,"d")]),Q("mi",null,"t"),Q("mo",null,"+"),Q("mn",null,"3"),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},")"),Q("mstyle",{scriptlevel:"0"},[Q("mspace",{width:"0.167em"})]),Q("mo",null,",")])],-1))]),Q("p",null,[t[211]||(t[211]=T("describes a simple harmonic oscillator, which is exactly soluble. Correspondingly, a response of ")),Q("mjx-container",k3,[(e(),a("svg",Z3,t[207]||(t[207]=[l('',1)]))),t[208]||(t[208]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"δ"),Q("mi",null,"x"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"t"),Q("mo",{stretchy:"false"},")")])],-1))]),t[212]||(t[212]=T(" at frequency ")),Q("mjx-container",b3,[(e(),a("svg",v3,t[209]||(t[209]=[l('',1)]))),t[210]||(t[210]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mn",null,"3"),Q("msub",null,[Q("mi",null,"ω"),Q("mi",null,"d")])])],-1))]),t[213]||(t[213]=T(" is observed. 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Limit cycles ​

We explain how HarmonicBalance.jl uses a new technique to find limit cycles in systems of nonlinear ODEs. For a more in depth overwiew see Chapter 6 in Jan Košata's PhD theses or del_Pino_2024.

Limit cycles from a Hopf bifurcation ​

',3)),t("p",null,[Q[2]||(Q[2]=T("The end product of the ")),Q[3]||(Q[3]=t("a",{href:"/HarmonicBalance.jl/v0.10.7/background/harmonic_balance#intro_hb"},"harmonic balance technique",-1)),Q[4]||(Q[4]=T(" are what we call the harmonic equations, i.e., first-order ODEs for the harmonic variables ")),t("mjx-container",m,[(e(),a("svg",d,Q[0]||(Q[0]=[s('',1)]))),Q[1]||(Q[1]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),Q[5]||(Q[5]=T(":"))]),t("mjx-container",n,[(e(),a("svg",r,Q[6]||(Q[6]=[s('',1)]))),Q[7]||(Q[7]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mfrac",null,[t("mrow",null,[t("mi",null,"d"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")]),t("mrow",null,[t("mi",null,"d"),t("mi",null,"T")])]),t("mo",null,"="),t("mover",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"G")]),t("mo",{accent:"true"},"―")]),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mo",{stretchy:"false"},")")])],-1))]),t("p",null,[Q[14]||(Q[14]=T("These Odes have no explicit time-dependence - they are autonomous. We have mostly been searching for steady states, which likewise show no time dependence. However, time-dependent solutions to autonomous ODEs can also exist. One mechanism for their creation is a ")),Q[15]||(Q[15]=t("a",{href:"https://en.wikipedia.org/wiki/Hopf_bifurcation",target:"_blank",rel:"noreferrer"},"Hopf bifurcation",-1)),Q[16]||(Q[16]=T(" - a critical point where a stable solution transitions into an unstable one. 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However, instead of evolving into another steady state, the system may assume a periodic orbit in phase space, giving a solution of the form"))]),t("mjx-container",x,[(e(),a("svg",w,Q[20]||(Q[20]=[s('',1)]))),Q[21]||(Q[21]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mn",null,"0")]),t("mo",null,"+"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-auto-op":"false"},"lc")])])]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mrow",{"data-mjx-texclass":"INNER"},[t("mo",{"data-mjx-texclass":"OPEN"},"("),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-auto-op":"false"},"lc")])])]),t("mi",null,"T"),t("mo",null,"+"),t("mi",null,"ϕ"),t("mo",{"data-mjx-texclass":"CLOSE"},")")])])],-1))]),Q[167]||(Q[167]=t("p",null,"which is an example of a limit cycle. We denote the originating steady state as Hopf-unstable.",-1)),t("p",null,[Q[40]||(Q[40]=T("We can continue to use harmonic balance as the solution still describes a harmonic response ")),Q[41]||(Q[41]=t("a",{href:"https://www.doi.org/10.1017/S0305004100054128",target:"_blank",rel:"noreferrer"},"Allwright (1977)",-1)),Q[42]||(Q[42]=T(". If we translate back to the the lab frame [variable ")),t("mjx-container",u,[(e(),a("svg",V,Q[22]||(Q[22]=[s('',1)]))),Q[23]||(Q[23]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),Q[43]||(Q[43]=T("], clearly, each frequency ")),t("mjx-container",L,[(e(),a("svg",f,Q[24]||(Q[24]=[s('',1)]))),Q[25]||(Q[25]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"j")])])],-1))]),Q[44]||(Q[44]=T(" constituting our harmonic ansatz [")),t("mjx-container",y,[(e(),a("svg",k,Q[26]||(Q[26]=[s('',1)]))),Q[27]||(Q[27]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),Q[45]||(Q[45]=T("], we obtain frequencies ")),t("mjx-container",M,[(e(),a("svg",Z,Q[28]||(Q[28]=[s('',1)]))),Q[29]||(Q[29]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"j")])])],-1))]),Q[46]||(Q[46]=T(" as well as ")),t("mjx-container",b,[(e(),a("svg",v,Q[30]||(Q[30]=[s('',1)]))),Q[31]||(Q[31]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"j")]),t("mo",null,"±"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])])])],-1))]),Q[47]||(Q[47]=T(" in the lab frame. Furthermore, as multiple harmonics now co-exist in the system, frequency conversion may take place, spawning further pairs ")),t("mjx-container",j,[(e(),a("svg",D,Q[32]||(Q[32]=[s('',1)]))),Q[33]||(Q[33]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"j")]),t("mo",null,"±"),t("mi",null,"k"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])])])],-1))]),Q[48]||(Q[48]=T(" with integer ")),t("mjx-container",O,[(e(),a("svg",C,Q[34]||(Q[34]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",style:{"stroke-width":"3"}})])])],-1)]))),Q[35]||(Q[35]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"k")])],-1))]),Q[49]||(Q[49]=T(". Therefore, to construct a harmonic ansatz capturing limit cycles, we simply add an integer number ")),t("mjx-container",R,[(e(),a("svg",E,Q[36]||(Q[36]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 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We again demonstrate by considering a single variable ")),t("mjx-container",G,[(e(),a("svg",J,Q[56]||(Q[56]=[s('',1)]))),Q[57]||(Q[57]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),Q[62]||(Q[62]=T(". We may try the simplest ansatz for a system driven at frequency ")),t("mjx-container",P,[(e(),a("svg",I,Q[58]||(Q[58]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),Q[59]||(Q[59]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),Q[63]||(Q[63]=T(","))]),t("mjx-container",F,[(e(),a("svg",q,Q[64]||(Q[64]=[s('',1)]))),Q[65]||(Q[65]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mi",null,"u"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msub",null,[t("mi",null,"v"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")"),t("mi",null,"sin"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),t("p",null,[Q[70]||(Q[70]=T("In this formulation, limit cycles may be obtained by solving the resulting harmonic equations with a Runge-Kutta type solver to obtain the time evolution of ")),t("mjx-container",K,[(e(),a("svg",W,Q[66]||(Q[66]=[s('',1)]))),Q[67]||(Q[67]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),Q[71]||(Q[71]=T(" and ")),t("mjx-container",U,[(e(),a("svg",$,Q[68]||(Q[68]=[s('',1)]))),Q[69]||(Q[69]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"v"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),Q[72]||(Q[72]=T(". See the ")),Q[73]||(Q[73]=t("a",{href:"/HarmonicBalance.jl/v0.10.7/tutorials/limit_cycles#limit_cycles"},"limit cycle tutorial",-1)),Q[74]||(Q[74]=T(" for an example."))]),Q[170]||(Q[170]=t("h3",{id:"Extended-ansatz",tabindex:"-1"},[T("Extended ansatz "),t("a",{class:"header-anchor",href:"#Extended-ansatz","aria-label":'Permalink to "Extended ansatz {#Extended-ansatz}"'},"​")],-1)),t("p",null,[Q[77]||(Q[77]=T("Including newly-emergent pairs of harmonics is in principle straightforward. Suppose a limit cycle has formed in our system with a frequency ")),t("mjx-container",Y,[(e(),a("svg",_,Q[75]||(Q[75]=[s('',1)]))),Q[76]||(Q[76]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-auto-op":"false"},"lc")])])])])],-1))]),Q[78]||(Q[78]=T(", prompting the ansatz"))]),t("mjx-container",t1,[(e(),a("svg",Q1,Q[79]||(Q[79]=[s('',1)]))),Q[80]||(Q[80]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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each of the ")),t("mjx-container",T1,[(e(),a("svg",a1,Q[81]||(Q[81]=[s('',1)]))),Q[82]||(Q[82]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω"),t("mo",null,"±"),t("mi",null,"k"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])])])],-1))]),Q[102]||(Q[102]=T(" pairs contributes 4 harmonic variables. The limit cycle frequency ")),t("mjx-container",e1,[(e(),a("svg",s1,Q[83]||(Q[83]=[s('',1)]))),Q[84]||(Q[84]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-auto-op":"false"},"lc")])])])])],-1))]),Q[103]||(Q[103]=T(" is also a variable in this formulation, but does not contribute a harmonic equation, since ")),t("mjx-container",l1,[(e(),a("svg",o1,Q[85]||(Q[85]=[s('',1)]))),Q[86]||(Q[86]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"d"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-auto-op":"false"},"lc")])])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mi",null,"d"),t("mi",null,"T"),t("mo",null,"="),t("mn",null,"0")])],-1))]),Q[104]||(Q[104]=T(" by construction. We thus arrive at a total of ")),t("mjx-container",m1,[(e(),a("svg",d1,Q[87]||(Q[87]=[s('',1)]))),Q[88]||(Q[88]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"4"),t("mi",null,"K")])],-1))]),Q[105]||(Q[105]=T(" harmonic equations in ")),t("mjx-container",n1,[(e(),a("svg",r1,Q[89]||(Q[89]=[s('',1)]))),Q[90]||(Q[90]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"4"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])],-1))]),Q[106]||(Q[106]=T(" variables. To obtain steady states, we must thus solve an underdetermined system, which has an infinite number of solutions. Given that we expect the limit cycles to possess ")),t("mjx-container",i1,[(e(),a("svg",h1,Q[91]||(Q[91]=[s('',1)]))),Q[92]||(Q[92]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"U"),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",{stretchy:"false"},")")])],-1))]),Q[107]||(Q[107]=T(" gauge freedom, this is a sensible observation. We may still use iterative numerical procedures such as the Newton method to find solutions one by one, but homotopy continuation is not applicable. In this formulation, steady staes states are characterised by zero entries for ")),t("mjx-container",p1,[(e(),a("svg",g1,Q[93]||(Q[93]=[s('',1)]))),Q[94]||(Q[94]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"2")]),t("mo",null,","),t("msub",null,[t("mi",null,"v"),t("mn",null,"2")]),t("mo",null,","),t("mo",null,"…"),t("msub",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])]),t("mo",null,","),t("msub",null,[t("mi",null,"v"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])])])],-1))]),Q[108]||(Q[108]=T(". The variable ")),t("mjx-container",H1,[(e(),a("svg",c1,Q[95]||(Q[95]=[s('',1)]))),Q[96]||(Q[96]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])])])],-1))]),Q[109]||(Q[109]=T(" is redundant and may take any value - the states therefore also appear infinitely degenerate, which, however, has no physical grounds. Oppositely, solutions may appear for which some of the limit cycle variables ")),t("mjx-container",x1,[(e(),a("svg",w1,Q[97]||(Q[97]=[s('',1)]))),Q[98]||(Q[98]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"2")]),t("mo",null,","),t("msub",null,[t("mi",null,"v"),t("mn",null,"2")]),t("mo",null,","),t("mo",null,"…"),t("msub",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])]),t("mo",null,","),t("msub",null,[t("mi",null,"v"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])])])],-1))]),Q[110]||(Q[110]=T(" are nonzero, but ")),t("mjx-container",u1,[(e(),a("svg",V1,Q[99]||(Q[99]=[s('',1)]))),Q[100]||(Q[100]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])]),t("mo",null,"="),t("mn",null,"0")])],-1))]),Q[111]||(Q[111]=T(". 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Discarding ")),t("mjx-container",G1,[(e(),a("svg",J1,Q[140]||(Q[140]=[s('',1)]))),Q[141]||(Q[141]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"2")])])],-1))]),Q[146]||(Q[146]=T(", we can therefore use ")),t("mjx-container",P1,[(e(),a("svg",I1,Q[142]||(Q[142]=[s('',1)]))),Q[143]||(Q[143]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"4"),t("mi",null,"K")])],-1))]),Q[147]||(Q[147]=T(" variables as our harmonic ansatz, i.e.,"))]),t("mjx-container",F1,[(e(),a("svg",q1,Q[148]||(Q[148]=[s('',1)]))),Q[149]||(Q[149]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mo",null,"="),t("mrow",{"data-mjx-texclass":"INNER"},[t("mo",{"data-mjx-texclass":"OPEN"},"("),t("mtable",{columnspacing:"1em",rowspacing:"4pt"},[t("mtr",null,[t("mtd",null,[t("msub",null,[t("mi",null,"u"),t("mn",null,"1")])])]),t("mtr",null,[t("mtd",null,[t("msub",null,[t("mi",null,"v"),t("mn",null,"1")])])]),t("mtr",null,[t("mtd",null,[t("msub",null,[t("mi",null,"v"),t("mn",null,"2")])])]),t("mtr",null,[t("mtd",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"⋮")])])]),t("mtr",null,[t("mtd",null,[t("msub",null,[t("mi",null,"v"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])])])]),t("mtr",null,[t("mtd",null,[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-auto-op":"false"},"lc")])])])])])]),t("mo",{"data-mjx-texclass":"CLOSE"},")")])])],-1))]),t("p",null,[Q[160]||(Q[160]=T("to remove the infinite degeneracy. Note that ")),t("mjx-container",K1,[(e(),a("svg",W1,Q[150]||(Q[150]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D719",d:"M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z",style:{"stroke-width":"3"}})])])],-1)]))),Q[151]||(Q[151]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ϕ")])],-1))]),Q[161]||(Q[161]=T(" is only defined modulo ")),t("mjx-container",U1,[(e(),a("svg",$1,Q[152]||(Q[152]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D70B",d:"M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z",style:{"stroke-width":"3"}})])])],-1)]))),Q[153]||(Q[153]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"π")])],-1))]),Q[162]||(Q[162]=T(", but its effect on the harmonic variables is not. Choosing ")),t("mjx-container",Y1,[(e(),a("svg",_1,Q[154]||(Q[154]=[s('',1)]))),Q[155]||(Q[155]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ϕ"),t("mo",null,"="),t("mo",null,"−"),t("mi",null,"arctan"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("msub",null,[t("mi",null,"u"),t("mn",null,"2")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("msub",null,[t("mi",null,"v"),t("mn",null,"2")]),t("mo",null,"+"),t("mi",null,"π")])],-1))]),Q[163]||(Q[163]=T(" would invert the signs of ")),t("mjx-container",t3,[(e(),a("svg",Q3,Q[156]||(Q[156]=[s('',1)]))),Q[157]||(Q[157]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"v"),t("mn",null,"2")]),t("mo",null,","),t("msub",null,[t("mi",null,"u"),t("mn",null,"3")]),t("mo",null,","),t("msub",null,[t("mi",null,"v"),t("mn",null,"3")])])],-1))]),Q[164]||(Q[164]=T(". As a result, each solution is doubly degenerate. Combined with the sign ambiguity of ")),t("mjx-container",T3,[(e(),a("svg",a3,Q[158]||(Q[158]=[s('',1)]))),Q[159]||(Q[159]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])])])],-1))]),Q[165]||(Q[165]=T(", we conclude that under the new ansatz, a limit cycle solution appears as a fourfold-degenerate steady state."))]),Q[173]||(Q[173]=t("p",null,[T("The harmonic equations can now be solved using homotopy continuation to obtain all steady states. 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Limit cycles ​

We explain how HarmonicBalance.jl uses a new technique to find limit cycles in systems of nonlinear ODEs. For a more in depth overwiew see Chapter 6 in Jan Košata's PhD theses or del_Pino_2024.

Limit cycles from a Hopf bifurcation ​

',3)),t("p",null,[Q[2]||(Q[2]=T("The end product of the ")),Q[3]||(Q[3]=t("a",{href:"/HarmonicBalance.jl/v0.10.7/background/harmonic_balance#intro_hb"},"harmonic balance technique",-1)),Q[4]||(Q[4]=T(" are what we call the harmonic equations, i.e., first-order ODEs for the harmonic variables ")),t("mjx-container",m,[(e(),a("svg",d,Q[0]||(Q[0]=[s('',1)]))),Q[1]||(Q[1]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),Q[5]||(Q[5]=T(":"))]),t("mjx-container",n,[(e(),a("svg",r,Q[6]||(Q[6]=[s('',1)]))),Q[7]||(Q[7]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mfrac",null,[t("mrow",null,[t("mi",null,"d"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")]),t("mrow",null,[t("mi",null,"d"),t("mi",null,"T")])]),t("mo",null,"="),t("mover",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"G")]),t("mo",{accent:"true"},"―")]),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mo",{stretchy:"false"},")")])],-1))]),t("p",null,[Q[14]||(Q[14]=T("These Odes have no explicit time-dependence - they are autonomous. We have mostly been searching for steady states, which likewise show no time dependence. However, time-dependent solutions to autonomous ODEs can also exist. One mechanism for their creation is a ")),Q[15]||(Q[15]=t("a",{href:"https://en.wikipedia.org/wiki/Hopf_bifurcation",target:"_blank",rel:"noreferrer"},"Hopf bifurcation",-1)),Q[16]||(Q[16]=T(" - a critical point where a stable solution transitions into an unstable one. 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We denote the originating steady state as Hopf-unstable.",-1)),t("p",null,[Q[40]||(Q[40]=T("We can continue to use harmonic balance as the solution still describes a harmonic response ")),Q[41]||(Q[41]=t("a",{href:"https://www.doi.org/10.1017/S0305004100054128",target:"_blank",rel:"noreferrer"},"Allwright (1977)",-1)),Q[42]||(Q[42]=T(". If we translate back to the the lab frame [variable ")),t("mjx-container",u,[(e(),a("svg",V,Q[22]||(Q[22]=[s('',1)]))),Q[23]||(Q[23]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),Q[43]||(Q[43]=T("], clearly, each frequency ")),t("mjx-container",L,[(e(),a("svg",f,Q[24]||(Q[24]=[s('',1)]))),Q[25]||(Q[25]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"j")])])],-1))]),Q[44]||(Q[44]=T(" constituting our harmonic ansatz [")),t("mjx-container",y,[(e(),a("svg",k,Q[26]||(Q[26]=[s('',1)]))),Q[27]||(Q[27]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"U")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),Q[45]||(Q[45]=T("], we obtain frequencies ")),t("mjx-container",M,[(e(),a("svg",Z,Q[28]||(Q[28]=[s('',1)]))),Q[29]||(Q[29]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"j")])])],-1))]),Q[46]||(Q[46]=T(" as well as ")),t("mjx-container",b,[(e(),a("svg",v,Q[30]||(Q[30]=[s('',1)]))),Q[31]||(Q[31]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"j")]),t("mo",null,"±"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])])])],-1))]),Q[47]||(Q[47]=T(" in the lab frame. Furthermore, as multiple harmonics now co-exist in the system, frequency conversion may take place, spawning further pairs ")),t("mjx-container",j,[(e(),a("svg",D,Q[32]||(Q[32]=[s('',1)]))),Q[33]||(Q[33]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"j")]),t("mo",null,"±"),t("mi",null,"k"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])])])],-1))]),Q[48]||(Q[48]=T(" with integer ")),t("mjx-container",O,[(e(),a("svg",C,Q[34]||(Q[34]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",style:{"stroke-width":"3"}})])])],-1)]))),Q[35]||(Q[35]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"k")])],-1))]),Q[49]||(Q[49]=T(". Therefore, to construct a harmonic ansatz capturing limit cycles, we simply add an integer number ")),t("mjx-container",R,[(e(),a("svg",E,Q[36]||(Q[36]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 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We again demonstrate by considering a single variable ")),t("mjx-container",G,[(e(),a("svg",J,Q[56]||(Q[56]=[s('',1)]))),Q[57]||(Q[57]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),Q[62]||(Q[62]=T(". We may try the simplest ansatz for a system driven at frequency ")),t("mjx-container",P,[(e(),a("svg",I,Q[58]||(Q[58]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),Q[59]||(Q[59]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),Q[63]||(Q[63]=T(","))]),t("mjx-container",F,[(e(),a("svg",q,Q[64]||(Q[64]=[s('',1)]))),Q[65]||(Q[65]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mi",null,"u"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msub",null,[t("mi",null,"v"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")"),t("mi",null,"sin"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),t("p",null,[Q[70]||(Q[70]=T("In this formulation, limit cycles may be obtained by solving the resulting harmonic equations with a Runge-Kutta type solver to obtain the time evolution of ")),t("mjx-container",K,[(e(),a("svg",W,Q[66]||(Q[66]=[s('',1)]))),Q[67]||(Q[67]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),Q[71]||(Q[71]=T(" and ")),t("mjx-container",U,[(e(),a("svg",$,Q[68]||(Q[68]=[s('',1)]))),Q[69]||(Q[69]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"v"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),Q[72]||(Q[72]=T(". See the ")),Q[73]||(Q[73]=t("a",{href:"/HarmonicBalance.jl/v0.10.7/tutorials/limit_cycles#limit_cycles"},"limit cycle tutorial",-1)),Q[74]||(Q[74]=T(" for an example."))]),Q[170]||(Q[170]=t("h3",{id:"Extended-ansatz",tabindex:"-1"},[T("Extended ansatz "),t("a",{class:"header-anchor",href:"#Extended-ansatz","aria-label":'Permalink to "Extended ansatz {#Extended-ansatz}"'},"​")],-1)),t("p",null,[Q[77]||(Q[77]=T("Including newly-emergent pairs of harmonics is in principle straightforward. Suppose a limit cycle has formed in our system with a frequency ")),t("mjx-container",Y,[(e(),a("svg",_,Q[75]||(Q[75]=[s('',1)]))),Q[76]||(Q[76]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-auto-op":"false"},"lc")])])])])],-1))]),Q[78]||(Q[78]=T(", prompting the ansatz"))]),t("mjx-container",t1,[(e(),a("svg",Q1,Q[79]||(Q[79]=[s('',1)]))),Q[80]||(Q[80]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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each of the ")),t("mjx-container",T1,[(e(),a("svg",a1,Q[81]||(Q[81]=[s('',1)]))),Q[82]||(Q[82]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω"),t("mo",null,"±"),t("mi",null,"k"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])])])],-1))]),Q[102]||(Q[102]=T(" pairs contributes 4 harmonic variables. The limit cycle frequency ")),t("mjx-container",e1,[(e(),a("svg",s1,Q[83]||(Q[83]=[s('',1)]))),Q[84]||(Q[84]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-auto-op":"false"},"lc")])])])])],-1))]),Q[103]||(Q[103]=T(" is also a variable in this formulation, but does not contribute a harmonic equation, since ")),t("mjx-container",l1,[(e(),a("svg",o1,Q[85]||(Q[85]=[s('',1)]))),Q[86]||(Q[86]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"d"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-auto-op":"false"},"lc")])])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mi",null,"d"),t("mi",null,"T"),t("mo",null,"="),t("mn",null,"0")])],-1))]),Q[104]||(Q[104]=T(" by construction. We thus arrive at a total of ")),t("mjx-container",m1,[(e(),a("svg",d1,Q[87]||(Q[87]=[s('',1)]))),Q[88]||(Q[88]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"4"),t("mi",null,"K")])],-1))]),Q[105]||(Q[105]=T(" harmonic equations in ")),t("mjx-container",n1,[(e(),a("svg",r1,Q[89]||(Q[89]=[s('',1)]))),Q[90]||(Q[90]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"4"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])],-1))]),Q[106]||(Q[106]=T(" variables. To obtain steady states, we must thus solve an underdetermined system, which has an infinite number of solutions. Given that we expect the limit cycles to possess ")),t("mjx-container",i1,[(e(),a("svg",h1,Q[91]||(Q[91]=[s('',1)]))),Q[92]||(Q[92]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"U"),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",{stretchy:"false"},")")])],-1))]),Q[107]||(Q[107]=T(" gauge freedom, this is a sensible observation. We may still use iterative numerical procedures such as the Newton method to find solutions one by one, but homotopy continuation is not applicable. In this formulation, steady staes states are characterised by zero entries for ")),t("mjx-container",p1,[(e(),a("svg",g1,Q[93]||(Q[93]=[s('',1)]))),Q[94]||(Q[94]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"2")]),t("mo",null,","),t("msub",null,[t("mi",null,"v"),t("mn",null,"2")]),t("mo",null,","),t("mo",null,"…"),t("msub",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])]),t("mo",null,","),t("msub",null,[t("mi",null,"v"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])])])],-1))]),Q[108]||(Q[108]=T(". The variable ")),t("mjx-container",H1,[(e(),a("svg",c1,Q[95]||(Q[95]=[s('',1)]))),Q[96]||(Q[96]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])])])],-1))]),Q[109]||(Q[109]=T(" is redundant and may take any value - the states therefore also appear infinitely degenerate, which, however, has no physical grounds. Oppositely, solutions may appear for which some of the limit cycle variables ")),t("mjx-container",x1,[(e(),a("svg",w1,Q[97]||(Q[97]=[s('',1)]))),Q[98]||(Q[98]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"2")]),t("mo",null,","),t("msub",null,[t("mi",null,"v"),t("mn",null,"2")]),t("mo",null,","),t("mo",null,"…"),t("msub",null,[t("mi",null,"u"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])]),t("mo",null,","),t("msub",null,[t("mi",null,"v"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2"),t("mi",null,"K"),t("mo",null,"+"),t("mn",null,"1")])])])],-1))]),Q[110]||(Q[110]=T(" are nonzero, but ")),t("mjx-container",u1,[(e(),a("svg",V1,Q[99]||(Q[99]=[s('',1)]))),Q[100]||(Q[100]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])]),t("mo",null,"="),t("mn",null,"0")])],-1))]),Q[111]||(Q[111]=T(". These violate our assumption of distinct harmonic variables corresponding to distinct frequencies and are therefore discarded."))]),Q[171]||(Q[171]=t("h3",{id:"gauge_fixing",tabindex:"-1"},[T("Gauge fixing "),t("a",{class:"header-anchor",href:"#gauge_fixing","aria-label":'Permalink to "Gauge fixing {#gauge_fixing}"'},"​")],-1)),t("p",null,[Q[120]||(Q[120]=T("We now constrain the system to remove the ")),t("mjx-container",L1,[(e(),a("svg",f1,Q[112]||(Q[112]=[s('',1)]))),Q[113]||(Q[113]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"U"),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",{stretchy:"false"},")")])],-1))]),Q[121]||(Q[121]=T(" gauge freedom. This is best done by explicitly writing out the free phase. Recall that our solution must be symmetric under a time translation symmetry, that is, taking ")),t("mjx-container",y1,[(e(),a("svg",k1,Q[114]||(Q[114]=[s('',1)]))),Q[115]||(Q[115]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"t"),t("mo",{stretchy:"false"},"→"),t("mi",null,"t"),t("mo",null,"+"),t("mn",null,"2"),t("mi",null,"π"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mi",null,"ω")])],-1))]),Q[122]||(Q[122]=T(". 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Choosing ")),t("mjx-container",Y1,[(e(),a("svg",_1,Q[154]||(Q[154]=[s('',1)]))),Q[155]||(Q[155]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ϕ"),t("mo",null,"="),t("mo",null,"−"),t("mi",null,"arctan"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("msub",null,[t("mi",null,"u"),t("mn",null,"2")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("msub",null,[t("mi",null,"v"),t("mn",null,"2")]),t("mo",null,"+"),t("mi",null,"π")])],-1))]),Q[163]||(Q[163]=T(" would invert the signs of ")),t("mjx-container",t3,[(e(),a("svg",Q3,Q[156]||(Q[156]=[s('',1)]))),Q[157]||(Q[157]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"v"),t("mn",null,"2")]),t("mo",null,","),t("msub",null,[t("mi",null,"u"),t("mn",null,"3")]),t("mo",null,","),t("msub",null,[t("mi",null,"v"),t("mn",null,"3")])])],-1))]),Q[164]||(Q[164]=T(". As a result, each solution is doubly degenerate. Combined with the sign ambiguity of ")),t("mjx-container",T3,[(e(),a("svg",a3,Q[158]||(Q[158]=[s('',1)]))),Q[159]||(Q[159]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"lc ")])])])],-1))]),Q[165]||(Q[165]=T(", we conclude that under the new ansatz, a limit cycle solution appears as a fourfold-degenerate steady state."))]),Q[173]||(Q[173]=t("p",null,[T("The harmonic equations can now be solved using homotopy continuation to obtain all steady states. 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To find this out, we insert the perturbation back into the harmonic ansatz. 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e="/HarmonicBalance.jl/v0.10.7/assets/ppwceys.DVQRnJSE.png",p="/HarmonicBalance.jl/v0.10.7/assets/mamyecy.2MzQm7AU.png",k="/HarmonicBalance.jl/v0.10.7/assets/lqzeigs.K-uWXBFJ.png",r="/HarmonicBalance.jl/v0.10.7/assets/czebnfy.B-Cc1T24.png",b=JSON.parse('{"title":"Parametric Pumping via Three-Wave Mixing","description":"","frontmatter":{},"headers":[],"relativePath":"examples/parametric_via_three_wave_mixing.md","filePath":"examples/parametric_via_three_wave_mixing.md"}'),d={name:"examples/parametric_via_three_wave_mixing.md"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},E={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.281ex",height:"2.034ex",role:"img",focusable:"false",viewBox:"0 -705 566 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F(u,i,C,x,v,D){return n(),a("div",null,[i[13]||(i[13]=l(`

Parametric Pumping via Three-Wave Mixing ​

julia
using HarmonicBalance, Plots
+using Plots.Measures
+using Random

System ​

julia
@variables β α ω ω0 F γ t x(t) # declare constant variables and a function x(t)
+diff_eq = DifferentialEquation(
+    d(x, t, 2) + ω0^2 * x + β * x^2 + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
+)
+add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)

1st order Krylov expansion ​

julia
harmonic_eq = get_krylov_equations(diff_eq; order=1)
+harmonic_eq.equations

If we both have quadratic and cubic nonlineariy, we observe the normal duffing oscillator response.

julia
varied = (ω => range(0.99, 1.1, 200)) # range of parameter values
+fixed = (α => 1.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025) # fixed parameters
+
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+plot(result; y="u1^2+v1^2")

If we set the cubic nonlinearity to zero, we recover the driven damped harmonic oscillator. Indeed, thefirst order the quadratic nonlinearity has no affect on the system.

julia
varied = (ω => range(0.99, 1.1, 100))
+fixed = (α => 0.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025)
+
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+plot(result; y="u1^2+v1^2")

2nd order Krylov expansion ​

',13)),s("p",null,[i[8]||(i[8]=t("The quadratic nonlinearity ")),s("mjx-container",g,[(n(),a("svg",E,i[0]||(i[0]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FD",d:"M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z",style:{"stroke-width":"3"}})])])],-1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"β")])],-1))]),i[9]||(i[9]=t(" together with the drive at 2ω gives the effective parametric drive ")),s("mjx-container",o,[(n(),a("svg",y,i[2]||(i[2]=[l('',1)]))),i[3]||(i[3]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"λ"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{"data-mjx-auto-op":"false"},"eff")])]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",null,"F"),s("mn",null,"1")]),s("mi",null,"β")]),s("mrow",null,[s("mn",null,"3"),s("mi",null,"m"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")])])])])],-1))]),i[10]||(i[10]=t(". But the cubic nonlinearity ")),s("mjx-container",T,[(n(),a("svg",Q,i[4]||(i[4]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),i[5]||(i[5]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"α")])],-1))]),i[11]||(i[11]=t(" is still needed to get the period doubling bifurcation through ")),s("mjx-container",m,[(n(),a("svg",c,i[6]||(i[6]=[l('',1)]))),i[7]||(i[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"λ"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{"data-mjx-auto-op":"false"},"eff")])])])],-1))]),i[12]||(i[12]=t("."))]),i[14]||(i[14]=l(`
julia
@variables β α ω ω0 F γ t x(t)
+diff_eq = DifferentialEquation(
+    d(x, t, 2) + ω0^2 * x + β * x^2 + α * x^3 + γ * d(x, t) ~ F * cos(2ω * t), x
+)
+
+add_harmonic!(diff_eq, x, ω)
+harmonic_eq2 = get_krylov_equations(diff_eq; order=2)
A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, ω0, F, β, α, γ
+
+Harmonic ansatz: 
+xˍt(t) = 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+(-(1//6)*F*v1(T)*β + (5//12)*(u1(T)^2)*v1(T)*(β^2) + (5//12)*(v1(T)^3)*(β^2) + (1//8)*v1(T)*(γ^2)*(ω^2) + (1//8)*v1(T)*(ω^4) - (1//4)*v1(T)*(ω^2)*(ω0^2) + (1//8)*v1(T)*(ω0^4) - (3//8)*(u1(T)^2)*v1(T)*α*(ω^2) + (3//8)*(u1(T)^2)*v1(T)*α*(ω0^2) - (3//8)*(v1(T)^3)*α*(ω^2) + (3//8)*(v1(T)^3)*α*(ω0^2) + (51//256)*(u1(T)^4)*v1(T)*(α^2) + (51//128)*(u1(T)^2)*(v1(T)^3)*(α^2) + (51//256)*(v1(T)^5)*(α^2)) / (ω^3) + (-(1//2)*u1(T)*γ*ω + (1//2)*v1(T)*(ω^2) - (1//2)*v1(T)*(ω0^2) - (3//8)*(u1(T)^2)*v1(T)*α - (3//8)*(v1(T)^3)*α) / ω ~ Differential(T)(u1(T))
+
+(-(1//6)*F*u1(T)*β - (5//12)*(u1(T)^3)*(β^2) - (5//12)*u1(T)*(v1(T)^2)*(β^2) - (1//8)*u1(T)*(γ^2)*(ω^2) - (1//8)*u1(T)*(ω^4) + (1//4)*u1(T)*(ω^2)*(ω0^2) - (1//8)*u1(T)*(ω0^4) + (3//8)*(u1(T)^3)*α*(ω^2) - (3//8)*(u1(T)^3)*α*(ω0^2) + (3//8)*u1(T)*(v1(T)^2)*α*(ω^2) - (3//8)*u1(T)*(v1(T)^2)*α*(ω0^2) - (51//256)*(u1(T)^5)*(α^2) - (51//128)*(u1(T)^3)*(v1(T)^2)*(α^2) - (51//256)*u1(T)*(v1(T)^4)*(α^2)) / (ω^3) + (-(1//2)*u1(T)*(ω^2) + (1//2)*u1(T)*(ω0^2) - (1//2)*v1(T)*γ*ω + (3//8)*(u1(T)^3)*α + (3//8)*u1(T)*(v1(T)^2)*α) / ω ~ Differential(T)(v1(T))
julia
varied = (ω => range(0.4, 1.1, 500))
+fixed = (α => 1.0, β => 2.0, ω0 => 1.0, γ => 0.001, F => 0.005)
+
+result = get_steady_states(harmonic_eq2, varied, fixed; threading=true)
+plot(result; y="v1")

julia
varied = (ω => range(0.4, 0.6, 100), F => range(1e-6, 0.01, 50))
+fixed = (α => 1.0, β => 2.0, ω0 => 1.0, γ => 0.01)
+
+result = get_steady_states(
+    harmonic_eq2, varied, fixed; threading=true, method=:total_degree
+)
+plot_phase_diagram(result; class="stable")

This page was generated using Literate.jl.

',8))])}const w=h(d,[["render",F]]);export{b as __pageData,w as default}; diff --git a/v0.10.7/assets/examples_parametric_via_three_wave_mixing.md.CT8h5xC1.lean.js b/v0.10.7/assets/examples_parametric_via_three_wave_mixing.md.CT8h5xC1.lean.js new file mode 100644 index 00000000..bdf2bbca --- /dev/null +++ b/v0.10.7/assets/examples_parametric_via_three_wave_mixing.md.CT8h5xC1.lean.js @@ -0,0 +1,44 @@ +import{_ as h,c as a,a4 as l,j as s,a as t,o as n}from"./chunks/framework.D3a-l5sU.js";const e="/HarmonicBalance.jl/v0.10.7/assets/ppwceys.DVQRnJSE.png",p="/HarmonicBalance.jl/v0.10.7/assets/mamyecy.2MzQm7AU.png",k="/HarmonicBalance.jl/v0.10.7/assets/lqzeigs.K-uWXBFJ.png",r="/HarmonicBalance.jl/v0.10.7/assets/czebnfy.B-Cc1T24.png",b=JSON.parse('{"title":"Parametric Pumping via Three-Wave Mixing","description":"","frontmatter":{},"headers":[],"relativePath":"examples/parametric_via_three_wave_mixing.md","filePath":"examples/parametric_via_three_wave_mixing.md"}'),d={name:"examples/parametric_via_three_wave_mixing.md"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},E={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.281ex",height:"2.034ex",role:"img",focusable:"false",viewBox:"0 -705 566 899","aria-hidden":"true"},o={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.889ex"},xmlns:"http://www.w3.org/2000/svg",width:"11.212ex",height:"3.096ex",role:"img",focusable:"false",viewBox:"0 -975.7 4955.8 1368.6","aria-hidden":"true"},T={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},Q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.448ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 640 453","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.357ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.302ex",height:"1.927ex",role:"img",focusable:"false",viewBox:"0 -694 1459.4 851.8","aria-hidden":"true"};function F(u,i,C,x,v,D){return n(),a("div",null,[i[13]||(i[13]=l(`

Parametric Pumping via Three-Wave Mixing ​

julia
using HarmonicBalance, Plots
+using Plots.Measures
+using Random

System ​

julia
@variables β α ω ω0 F γ t x(t) # declare constant variables and a function x(t)
+diff_eq = DifferentialEquation(
+    d(x, t, 2) + ω0^2 * x + β * x^2 + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
+)
+add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)

1st order Krylov expansion ​

julia
harmonic_eq = get_krylov_equations(diff_eq; order=1)
+harmonic_eq.equations

If we both have quadratic and cubic nonlineariy, we observe the normal duffing oscillator response.

julia
varied = (ω => range(0.99, 1.1, 200)) # range of parameter values
+fixed = (α => 1.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025) # fixed parameters
+
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+plot(result; y="u1^2+v1^2")

If we set the cubic nonlinearity to zero, we recover the driven damped harmonic oscillator. Indeed, thefirst order the quadratic nonlinearity has no affect on the system.

julia
varied = (ω => range(0.99, 1.1, 100))
+fixed = (α => 0.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025)
+
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+plot(result; y="u1^2+v1^2")

2nd order Krylov expansion ​

',13)),s("p",null,[i[8]||(i[8]=t("The quadratic nonlinearity ")),s("mjx-container",g,[(n(),a("svg",E,i[0]||(i[0]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FD",d:"M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z",style:{"stroke-width":"3"}})])])],-1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"β")])],-1))]),i[9]||(i[9]=t(" together with the drive at 2ω gives the effective parametric drive ")),s("mjx-container",o,[(n(),a("svg",y,i[2]||(i[2]=[l('',1)]))),i[3]||(i[3]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"λ"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{"data-mjx-auto-op":"false"},"eff")])]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",null,"F"),s("mn",null,"1")]),s("mi",null,"β")]),s("mrow",null,[s("mn",null,"3"),s("mi",null,"m"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")])])])])],-1))]),i[10]||(i[10]=t(". But the cubic nonlinearity ")),s("mjx-container",T,[(n(),a("svg",Q,i[4]||(i[4]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),i[5]||(i[5]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"α")])],-1))]),i[11]||(i[11]=t(" is still needed to get the period doubling bifurcation through ")),s("mjx-container",m,[(n(),a("svg",c,i[6]||(i[6]=[l('',1)]))),i[7]||(i[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"λ"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{"data-mjx-auto-op":"false"},"eff")])])])],-1))]),i[12]||(i[12]=t("."))]),i[14]||(i[14]=l(`
julia
@variables β α ω ω0 F γ t x(t)
+diff_eq = DifferentialEquation(
+    d(x, t, 2) + ω0^2 * x + β * x^2 + α * x^3 + γ * d(x, t) ~ F * cos(2ω * t), x
+)
+
+add_harmonic!(diff_eq, x, ω)
+harmonic_eq2 = get_krylov_equations(diff_eq; order=2)
A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, ω0, F, β, α, γ
+
+Harmonic ansatz: 
+xˍt(t) = 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+(-(1//6)*F*v1(T)*β + (5//12)*(u1(T)^2)*v1(T)*(β^2) + (5//12)*(v1(T)^3)*(β^2) + (1//8)*v1(T)*(γ^2)*(ω^2) + (1//8)*v1(T)*(ω^4) - (1//4)*v1(T)*(ω^2)*(ω0^2) + (1//8)*v1(T)*(ω0^4) - (3//8)*(u1(T)^2)*v1(T)*α*(ω^2) + (3//8)*(u1(T)^2)*v1(T)*α*(ω0^2) - (3//8)*(v1(T)^3)*α*(ω^2) + (3//8)*(v1(T)^3)*α*(ω0^2) + (51//256)*(u1(T)^4)*v1(T)*(α^2) + (51//128)*(u1(T)^2)*(v1(T)^3)*(α^2) + (51//256)*(v1(T)^5)*(α^2)) / (ω^3) + (-(1//2)*u1(T)*γ*ω + (1//2)*v1(T)*(ω^2) - (1//2)*v1(T)*(ω0^2) - (3//8)*(u1(T)^2)*v1(T)*α - (3//8)*(v1(T)^3)*α) / ω ~ Differential(T)(u1(T))
+
+(-(1//6)*F*u1(T)*β - (5//12)*(u1(T)^3)*(β^2) - (5//12)*u1(T)*(v1(T)^2)*(β^2) - (1//8)*u1(T)*(γ^2)*(ω^2) - (1//8)*u1(T)*(ω^4) + (1//4)*u1(T)*(ω^2)*(ω0^2) - (1//8)*u1(T)*(ω0^4) + (3//8)*(u1(T)^3)*α*(ω^2) - (3//8)*(u1(T)^3)*α*(ω0^2) + (3//8)*u1(T)*(v1(T)^2)*α*(ω^2) - (3//8)*u1(T)*(v1(T)^2)*α*(ω0^2) - (51//256)*(u1(T)^5)*(α^2) - (51//128)*(u1(T)^3)*(v1(T)^2)*(α^2) - (51//256)*u1(T)*(v1(T)^4)*(α^2)) / (ω^3) + (-(1//2)*u1(T)*(ω^2) + (1//2)*u1(T)*(ω0^2) - (1//2)*v1(T)*γ*ω + (3//8)*(u1(T)^3)*α + (3//8)*u1(T)*(v1(T)^2)*α) / ω ~ Differential(T)(v1(T))
julia
varied = (ω => range(0.4, 1.1, 500))
+fixed = (α => 1.0, β => 2.0, ω0 => 1.0, γ => 0.001, F => 0.005)
+
+result = get_steady_states(harmonic_eq2, varied, fixed; threading=true)
+plot(result; y="v1")

julia
varied = (ω => range(0.4, 0.6, 100), F => range(1e-6, 0.01, 50))
+fixed = (α => 1.0, β => 2.0, ω0 => 1.0, γ => 0.01)
+
+result = get_steady_states(
+    harmonic_eq2, varied, fixed; threading=true, method=:total_degree
+)
+plot_phase_diagram(result; class="stable")

This page was generated using Literate.jl.

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454","aria-hidden":"true"},O={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},R={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"};function S(N,a,P,z,G,K){return n(),e("div",null,[a[66]||(a[66]=t("h1",{id:"parametron",tabindex:"-1"},[s("Parametrically driven resonator "),t("a",{class:"header-anchor",href:"#parametron","aria-label":'Permalink to "Parametrically driven resonator {#parametron}"'},"​")],-1)),a[67]||(a[67]=t("p",null,[s("One of the most famous effects displaced by nonlinear oscillators is parametric resonance, where the frequency of the linear resonator is modulated in time "),t("a",{href:"https://doi.org/10.1103/PhysRevE.94.022201",target:"_blank",rel:"noreferrer"},"Phys. Rev. E 94, 022201 (2016)"),s(". In the following we analyse this system, governed by the equations")],-1)),t("mjx-container",h,[(n(),e("svg",m,a[0]||(a[0]=[i('',1)]))),a[1]||(a[1]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"γ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msup",null,[t("mi",{mathvariant:"normal"},"Ω"),t("mn",null,"2")]),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",null,"−"),t("mi",null,"λ"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"2"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("mi",null,"ψ"),t("mo",{stretchy:"false"},")"),t("mo",{stretchy:"false"},")"),t("mi",null,"x"),t("mo",null,"+"),t("mi",null,"α"),t("msup",null,[t("mi",null,"x"),t("mn",null,"3")]),t("mo",null,"+"),t("mi",null,"η"),t("msup",null,[t("mi",null,"x"),t("mn",null,"2")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",null,"+"),t("msub",null,[t("mi",null,"F"),t("mtext",null,"d")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mn",null,"0")])],-1))]),t("p",null,[a[6]||(a[6]=s("where for completeness we also considered an external drive term ")),t("mjx-container",g,[(n(),e("svg",k,a[2]||(a[2]=[i('',1)]))),a[3]||(a[3]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mtext",null,"d")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"F"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("mi",null,"θ"),t("mo",{stretchy:"false"},")")])],-1))]),a[7]||(a[7]=s(" and a nonlinear damping term ")),t("mjx-container",c,[(n(),e("svg",u,a[4]||(a[4]=[i('',1)]))),a[5]||(a[5]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"η"),t("msup",null,[t("mi",null,"x"),t("mn",null,"2")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])])])],-1))])]),a[68]||(a[68]=i('

To implement this system in Harmonic Balance, we first import the library

julia
using HarmonicBalance
',2)),t("p",null,[a[10]||(a[10]=s("Subsequently, we type define parameters in the problem and the oscillating amplitude function ")),t("mjx-container",y,[(n(),e("svg",H,a[8]||(a[8]=[i('',1)]))),a[9]||(a[9]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[11]||(a[11]=s(" using the ")),a[12]||(a[12]=t("code",null,"variables",-1)),a[13]||(a[13]=s(" macro from ")),a[14]||(a[14]=t("code",null,"Symbolics.jl",-1))]),a[69]||(a[69]=i(`
julia
@variables ω₀ γ λ F η α ω t x(t)
+
+natural_equation =
+    d(d(x, t), t) +
+    γ * d(x, t) +
+    (ω₀^2 - λ * cos(2 * ω * t)) * x +
+    α * x^3 +
+    η * d(x, t) * x^2
+forces = F * cos(ω * t)
+diff_eq = DifferentialEquation(natural_equation + forces, x)
System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ;   
+
+Differential(t)(Differential(t)(x(t))) + F*cos(t*ω) + Differential(t)(x(t))*γ + x(t)*(-cos(2t*ω)*λ + ω₀^2) + (x(t)^3)*α + (x(t)^2)*Differential(t)(x(t))*η ~ 0

Note that an equation of the form

`,3)),t("mjx-container",x,[(n(),e("svg",w,a[15]||(a[15]=[i('',1)]))),a[16]||(a[16]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"m"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mo",null,"+"),t("mi",null,"m"),t("msubsup",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"0")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2")])]),t("mrow",{"data-mjx-texclass":"INNER"},[t("mo",{"data-mjx-texclass":"OPEN"},"("),t("mn",null,"1"),t("mo",null,"−"),t("mi",null,"λ"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"2"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("mi",null,"ψ"),t("mo",{stretchy:"false"},")"),t("mo",{"data-mjx-texclass":"CLOSE"},")")]),t("mi",null,"x"),t("mo",null,"+"),t("mi",null,"γ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",null,"+"),t("mi",null,"α"),t("msup",null,[t("mi",null,"x"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"3")])]),t("mo",null,"+"),t("mi",null,"η"),t("msup",null,[t("mi",null,"x"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2")])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",null,"="),t("mi",null,"F"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mi",null,"ω"),t("mi",null,"t")])],-1))]),a[70]||(a[70]=t("p",null,[s("can be brought to dimensionless form by rescaling the units as described in "),t("a",{href:"https://doi.org/10.1103/PhysRevE.94.022201",target:"_blank",rel:"noreferrer"},"Phys. Rev. E 94, 022201 (2016)"),s(".")],-1)),t("p",null,[a[23]||(a[23]=s("We are interested in studying the response of the oscillator to parametric driving and forcing. In particular, we focus on the first parametric resonance of the system, i.e. operating around twice the bare frequency of the undriven oscillator ")),t("mjx-container",f,[(n(),e("svg",E,a[17]||(a[17]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[18]||(a[18]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[24]||(a[24]=s(" while the frequency of the external drive is also ")),t("mjx-container",L,[(n(),e("svg",b,a[19]||(a[19]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[20]||(a[20]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[25]||(a[25]=s(". For this purpose, we consider a harmonic ansatz which contains a single frequency: ")),t("mjx-container",v,[(n(),e("svg",F,a[21]||(a[21]=[i('',1)]))),a[22]||(a[22]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"≈"),t("mi",null,"u"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"v"),t("mi",null,"sin"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[26]||(a[26]=s(". In HarmonicBalance, we can do this via ")),a[27]||(a[27]=t("code",null,"add_harmonic",-1)),a[28]||(a[28]=s(" command:"))]),a[71]||(a[71]=i(`
julia
add_harmonic!(diff_eq, x, ω);

and replacing this by the time independent (averaged) equations of motion. This can be simply done by writing

julia
harmonic_eq = get_harmonic_equations(diff_eq)
A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω₀, λ, F, η
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+F - (1//2)*u1(T)*λ + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω₀^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*Differential(T)(u1(T))*η + (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*η + (3//4)*u1(T)*(v1(T)^2)*α + (1//4)*(v1(T)^2)*Differential(T)(u1(T))*η + (1//4)*(u1(T)^2)*v1(T)*η*ω + (1//4)*(v1(T)^3)*η*ω ~ 0
+
+Differential(T)(v1(T))*γ + (1//2)*v1(T)*λ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω₀^2) + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*η + (3//4)*(u1(T)^2)*v1(T)*α + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*η + (3//4)*Differential(T)(v1(T))*(v1(T)^2)*η + (3//4)*(v1(T)^3)*α - (1//4)*(u1(T)^3)*η*ω - (1//4)*u1(T)*(v1(T)^2)*η*ω ~ 0

The output of these equations are consistent with the result found in the literature. Now we are interested in the linear response spectrum, which we can obtain from the solutions to the averaged equations (rotating frame) as a function of the external drive, after fixing all other parameters in the system. A call to get_steady_states then retrieves all steadystates found allong the sweep employing the homotopy continuation method, which occurs in a complex space (see the nice HomotopyContinuation.jl docs)

1D parameters ​

`,6)),t("p",null,[a[31]||(a[31]=s("We start with a ")),a[32]||(a[32]=t("code",null,"varied",-1)),a[33]||(a[33]=s(" set containing one parameter, ")),t("mjx-container",M,[(n(),e("svg",V,a[29]||(a[29]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[30]||(a[30]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[34]||(a[34]=s(","))]),a[72]||(a[72]=i(`
julia
fixed = (ω₀ => 1.0, γ => 1e-2, λ => 5e-2, F => 1e-3, α => 1.0, η => 0.3)
+varied = ω => range(0.9, 1.1, 100)
+
+result = get_steady_states(harmonic_eq, varied, fixed)
A steady state result for 100 parameter points
+
+Solution branches:   5
+   of which real:    5
+   of which stable:  3
+
+Classes: stable, physical, Hopf, binary_labels
`,2)),t("p",null,[a[37]||(a[37]=s("In ")),a[38]||(a[38]=t("code",null,"get_steady_states",-1)),a[39]||(a[39]=s(", the default value for the keyword ")),a[40]||(a[40]=t("code",null,"method=:random_warmup",-1)),a[41]||(a[41]=s(" initiates the homotopy in a generalised version of the harmonic equations, where parameters become random complex numbers. A parameter homotopy then follows to each of the frequency values ")),t("mjx-container",D,[(n(),e("svg",C,a[35]||(a[35]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[36]||(a[36]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[42]||(a[42]=s(" in sweep. This offers speed-up, but requires to be tested in each scenario againts the method ")),a[43]||(a[43]=t("code",null,":total_degree",-1)),a[44]||(a[44]=s(", which initializes the homotopy in a total degree system (maximum number of roots), but needs to track significantly more homotopy paths and there is slower. The ")),a[45]||(a[45]=t("code",null,"threading",-1)),a[46]||(a[46]=s(" keyword enables parallel tracking of homotopy paths, and it's set to ")),a[47]||(a[47]=t("code",null,"false",-1)),a[48]||(a[48]=s(" simply because we are using a single core computer for now."))]),a[73]||(a[73]=i(`

After solving the system, we can save the full output of the simulation and the model (e.g. symbolic expressions for the harmonic equations) into a file

julia
HarmonicBalance.save("parametron_result.jld2", result);
┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(identity).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(identity).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631

During the execution of get_steady_states, different solution branches are classified by their proximity in complex space, with subsequent filtering of real (physically accceptable solutions). In addition, the stability properties of each steady state is assesed from the eigenvalues of the Jacobian matrix. All this information can be succintly represented in a 1D plot via

julia
plot(result; x="ω", y="sqrt(u1^2 + v1^2)")

The user can also introduce custom clases based on parameter conditions via classify_solutions!. Plots can be overlaid and use keywords from Plots, MarkdownAST.LineBreak()

julia
classify_solutions!(result, "sqrt(u1^2 + v1^2) > 0.1", "large")
+plot(result, "sqrt(u1^2 + v1^2)"; class=["physical", "large"], style=:dash)
+plot!(result, "sqrt(u1^2 + v1^2)"; not_class="large")

Alternatively, we may visualise all underlying solutions, including complex ones,

julia
plot(result, "sqrt(u1^2 + v1^2)"; class="all")

2D parameters ​

',13)),t("p",null,[a[53]||(a[53]=s(`The parametrically driven oscillator boasts a stability diagram called "Arnold's tongues" delineating zones where the oscillator is stable from those where it is exponentially unstable (if the nonlinearity was absence). We can retrieve this diagram by calculating the steady states as a function of external detuning `)),t("mjx-container",Z,[(n(),e("svg",j,a[49]||(a[49]=[i('',1)]))),a[50]||(a[50]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"δ"),t("mo",null,"="),t("msub",null,[t("mi",null,"ω"),t("mi",null,"L")]),t("mo",null,"−"),t("msub",null,[t("mi",null,"ω"),t("mn",null,"0")])])],-1))]),a[54]||(a[54]=s(" and the parametric drive strength ")),t("mjx-container",B,[(n(),e("svg",A,a[51]||(a[51]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),a[52]||(a[52]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"λ")])],-1))]),a[55]||(a[55]=s("."))]),t("p",null,[a[60]||(a[60]=s("To perform a 2D sweep over driving frequency ")),t("mjx-container",q,[(n(),e("svg",J,a[56]||(a[56]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[57]||(a[57]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[61]||(a[61]=s(" and parametric drive strength ")),t("mjx-container",O,[(n(),e("svg",R,a[58]||(a[58]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),a[59]||(a[59]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"λ")])],-1))]),a[62]||(a[62]=s(", we keep ")),a[63]||(a[63]=t("code",null,"fixed",-1)),a[64]||(a[64]=s(" from before but include 2 variables in ")),a[65]||(a[65]=t("code",null,"varied",-1))]),a[74]||(a[74]=i(`
julia
varied = (ω => range(0.8, 1.2, 50), λ => range(0.001, 0.6, 50))
+result_2D = get_steady_states(harmonic_eq, varied, fixed);

+Solving for 2500 parameters...  53%|██████████▋         |  ETA: 0:00:01\x1B[K
+  # parameters solved:  1328\x1B[K
+  # paths tracked:      6640\x1B[K
+\x1B[A
+\x1B[A
+
+
+\x1B[K\x1B[A
+\x1B[K\x1B[A
+Solving for 2500 parameters...  83%|████████████████▋   |  ETA: 0:00:00\x1B[K
+  # parameters solved:  2085\x1B[K
+  # paths tracked:      10425\x1B[K
+\x1B[A
+\x1B[A
+
+
+\x1B[K\x1B[A
+\x1B[K\x1B[A
+Solving for 2500 parameters... 100%|████████████████████| Time: 0:00:01\x1B[K
+  # parameters solved:  2500\x1B[K
+  # paths tracked:      12500\x1B[K

Now, we count the number of solutions for each point and represent the corresponding phase diagram in parameter space. This is done using plot_phase_diagram. Only counting stable solutions,

julia
plot_phase_diagram(result_2D; class="stable")

In addition to phase diagrams, we can plot functions of the solution. The syntax is identical to 1D plotting. Let us overlay 2 branches into a single plot,

julia
# overlay branches with different colors
+plot(result_2D, "sqrt(u1^2 + v1^2)"; branch=1, class="stable", camera=(60, -40))
+plot!(result_2D, "sqrt(u1^2 + v1^2)"; branch=2, class="stable", color=:red)

Note that solutions are ordered in parameter space according to their closest neighbors. Plots can again be limited to a given class (e.g stable solutions only) through the keyword argument class.

This page was generated using Literate.jl.

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454","aria-hidden":"true"},O={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},R={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"};function S(N,a,P,z,G,K){return n(),e("div",null,[a[66]||(a[66]=t("h1",{id:"parametron",tabindex:"-1"},[s("Parametrically driven resonator "),t("a",{class:"header-anchor",href:"#parametron","aria-label":'Permalink to "Parametrically driven resonator {#parametron}"'},"​")],-1)),a[67]||(a[67]=t("p",null,[s("One of the most famous effects displaced by nonlinear oscillators is parametric resonance, where the frequency of the linear resonator is modulated in time "),t("a",{href:"https://doi.org/10.1103/PhysRevE.94.022201",target:"_blank",rel:"noreferrer"},"Phys. Rev. E 94, 022201 (2016)"),s(". In the following we analyse this system, governed by the equations")],-1)),t("mjx-container",h,[(n(),e("svg",m,a[0]||(a[0]=[i('',1)]))),a[1]||(a[1]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"γ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msup",null,[t("mi",{mathvariant:"normal"},"Ω"),t("mn",null,"2")]),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",null,"−"),t("mi",null,"λ"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"2"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("mi",null,"ψ"),t("mo",{stretchy:"false"},")"),t("mo",{stretchy:"false"},")"),t("mi",null,"x"),t("mo",null,"+"),t("mi",null,"α"),t("msup",null,[t("mi",null,"x"),t("mn",null,"3")]),t("mo",null,"+"),t("mi",null,"η"),t("msup",null,[t("mi",null,"x"),t("mn",null,"2")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",null,"+"),t("msub",null,[t("mi",null,"F"),t("mtext",null,"d")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mn",null,"0")])],-1))]),t("p",null,[a[6]||(a[6]=s("where for completeness we also considered an external drive term ")),t("mjx-container",g,[(n(),e("svg",k,a[2]||(a[2]=[i('',1)]))),a[3]||(a[3]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mtext",null,"d")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"F"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("mi",null,"θ"),t("mo",{stretchy:"false"},")")])],-1))]),a[7]||(a[7]=s(" and a nonlinear damping term ")),t("mjx-container",c,[(n(),e("svg",u,a[4]||(a[4]=[i('',1)]))),a[5]||(a[5]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"η"),t("msup",null,[t("mi",null,"x"),t("mn",null,"2")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])])])],-1))])]),a[68]||(a[68]=i('

To implement this system in Harmonic Balance, we first import the library

julia
using HarmonicBalance
',2)),t("p",null,[a[10]||(a[10]=s("Subsequently, we type define parameters in the problem and the oscillating amplitude function ")),t("mjx-container",y,[(n(),e("svg",H,a[8]||(a[8]=[i('',1)]))),a[9]||(a[9]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[11]||(a[11]=s(" using the ")),a[12]||(a[12]=t("code",null,"variables",-1)),a[13]||(a[13]=s(" macro from ")),a[14]||(a[14]=t("code",null,"Symbolics.jl",-1))]),a[69]||(a[69]=i(`
julia
@variables ω₀ γ λ F η α ω t x(t)
+
+natural_equation =
+    d(d(x, t), t) +
+    γ * d(x, t) +
+    (ω₀^2 - λ * cos(2 * ω * t)) * x +
+    α * x^3 +
+    η * d(x, t) * x^2
+forces = F * cos(ω * t)
+diff_eq = DifferentialEquation(natural_equation + forces, x)
System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ;   
+
+Differential(t)(Differential(t)(x(t))) + F*cos(t*ω) + Differential(t)(x(t))*γ + x(t)*(-cos(2t*ω)*λ + ω₀^2) + (x(t)^3)*α + (x(t)^2)*Differential(t)(x(t))*η ~ 0

Note that an equation of the form

`,3)),t("mjx-container",x,[(n(),e("svg",w,a[15]||(a[15]=[i('',1)]))),a[16]||(a[16]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"m"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mo",null,"+"),t("mi",null,"m"),t("msubsup",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"0")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2")])]),t("mrow",{"data-mjx-texclass":"INNER"},[t("mo",{"data-mjx-texclass":"OPEN"},"("),t("mn",null,"1"),t("mo",null,"−"),t("mi",null,"λ"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"2"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("mi",null,"ψ"),t("mo",{stretchy:"false"},")"),t("mo",{"data-mjx-texclass":"CLOSE"},")")]),t("mi",null,"x"),t("mo",null,"+"),t("mi",null,"γ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",null,"+"),t("mi",null,"α"),t("msup",null,[t("mi",null,"x"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"3")])]),t("mo",null,"+"),t("mi",null,"η"),t("msup",null,[t("mi",null,"x"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2")])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",null,"="),t("mi",null,"F"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mi",null,"ω"),t("mi",null,"t")])],-1))]),a[70]||(a[70]=t("p",null,[s("can be brought to dimensionless form by rescaling the units as described in "),t("a",{href:"https://doi.org/10.1103/PhysRevE.94.022201",target:"_blank",rel:"noreferrer"},"Phys. Rev. E 94, 022201 (2016)"),s(".")],-1)),t("p",null,[a[23]||(a[23]=s("We are interested in studying the response of the oscillator to parametric driving and forcing. In particular, we focus on the first parametric resonance of the system, i.e. operating around twice the bare frequency of the undriven oscillator ")),t("mjx-container",f,[(n(),e("svg",E,a[17]||(a[17]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[18]||(a[18]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[24]||(a[24]=s(" while the frequency of the external drive is also ")),t("mjx-container",L,[(n(),e("svg",b,a[19]||(a[19]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[20]||(a[20]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[25]||(a[25]=s(". For this purpose, we consider a harmonic ansatz which contains a single frequency: ")),t("mjx-container",v,[(n(),e("svg",F,a[21]||(a[21]=[i('',1)]))),a[22]||(a[22]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"≈"),t("mi",null,"u"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"v"),t("mi",null,"sin"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[26]||(a[26]=s(". In HarmonicBalance, we can do this via ")),a[27]||(a[27]=t("code",null,"add_harmonic",-1)),a[28]||(a[28]=s(" command:"))]),a[71]||(a[71]=i(`
julia
add_harmonic!(diff_eq, x, ω);

and replacing this by the time independent (averaged) equations of motion. This can be simply done by writing

julia
harmonic_eq = get_harmonic_equations(diff_eq)
A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω₀, λ, F, η
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+F - (1//2)*u1(T)*λ + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω₀^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*Differential(T)(u1(T))*η + (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*η + (3//4)*u1(T)*(v1(T)^2)*α + (1//4)*(v1(T)^2)*Differential(T)(u1(T))*η + (1//4)*(u1(T)^2)*v1(T)*η*ω + (1//4)*(v1(T)^3)*η*ω ~ 0
+
+Differential(T)(v1(T))*γ + (1//2)*v1(T)*λ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω₀^2) + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*η + (3//4)*(u1(T)^2)*v1(T)*α + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*η + (3//4)*Differential(T)(v1(T))*(v1(T)^2)*η + (3//4)*(v1(T)^3)*α - (1//4)*(u1(T)^3)*η*ω - (1//4)*u1(T)*(v1(T)^2)*η*ω ~ 0

The output of these equations are consistent with the result found in the literature. Now we are interested in the linear response spectrum, which we can obtain from the solutions to the averaged equations (rotating frame) as a function of the external drive, after fixing all other parameters in the system. A call to get_steady_states then retrieves all steadystates found allong the sweep employing the homotopy continuation method, which occurs in a complex space (see the nice HomotopyContinuation.jl docs)

1D parameters ​

`,6)),t("p",null,[a[31]||(a[31]=s("We start with a ")),a[32]||(a[32]=t("code",null,"varied",-1)),a[33]||(a[33]=s(" set containing one parameter, ")),t("mjx-container",M,[(n(),e("svg",V,a[29]||(a[29]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[30]||(a[30]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[34]||(a[34]=s(","))]),a[72]||(a[72]=i(`
julia
fixed = (ω₀ => 1.0, γ => 1e-2, λ => 5e-2, F => 1e-3, α => 1.0, η => 0.3)
+varied = ω => range(0.9, 1.1, 100)
+
+result = get_steady_states(harmonic_eq, varied, fixed)
A steady state result for 100 parameter points
+
+Solution branches:   5
+   of which real:    5
+   of which stable:  3
+
+Classes: stable, physical, Hopf, binary_labels
`,2)),t("p",null,[a[37]||(a[37]=s("In ")),a[38]||(a[38]=t("code",null,"get_steady_states",-1)),a[39]||(a[39]=s(", the default value for the keyword ")),a[40]||(a[40]=t("code",null,"method=:random_warmup",-1)),a[41]||(a[41]=s(" initiates the homotopy in a generalised version of the harmonic equations, where parameters become random complex numbers. A parameter homotopy then follows to each of the frequency values ")),t("mjx-container",D,[(n(),e("svg",C,a[35]||(a[35]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[36]||(a[36]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[42]||(a[42]=s(" in sweep. This offers speed-up, but requires to be tested in each scenario againts the method ")),a[43]||(a[43]=t("code",null,":total_degree",-1)),a[44]||(a[44]=s(", which initializes the homotopy in a total degree system (maximum number of roots), but needs to track significantly more homotopy paths and there is slower. The ")),a[45]||(a[45]=t("code",null,"threading",-1)),a[46]||(a[46]=s(" keyword enables parallel tracking of homotopy paths, and it's set to ")),a[47]||(a[47]=t("code",null,"false",-1)),a[48]||(a[48]=s(" simply because we are using a single core computer for now."))]),a[73]||(a[73]=i(`

After solving the system, we can save the full output of the simulation and the model (e.g. symbolic expressions for the harmonic equations) into a file

julia
HarmonicBalance.save("parametron_result.jld2", result);
┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(identity).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(identity).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631

During the execution of get_steady_states, different solution branches are classified by their proximity in complex space, with subsequent filtering of real (physically accceptable solutions). In addition, the stability properties of each steady state is assesed from the eigenvalues of the Jacobian matrix. All this information can be succintly represented in a 1D plot via

julia
plot(result; x="ω", y="sqrt(u1^2 + v1^2)")

The user can also introduce custom clases based on parameter conditions via classify_solutions!. Plots can be overlaid and use keywords from Plots, MarkdownAST.LineBreak()

julia
classify_solutions!(result, "sqrt(u1^2 + v1^2) > 0.1", "large")
+plot(result, "sqrt(u1^2 + v1^2)"; class=["physical", "large"], style=:dash)
+plot!(result, "sqrt(u1^2 + v1^2)"; not_class="large")

Alternatively, we may visualise all underlying solutions, including complex ones,

julia
plot(result, "sqrt(u1^2 + v1^2)"; class="all")

2D parameters ​

',13)),t("p",null,[a[53]||(a[53]=s(`The parametrically driven oscillator boasts a stability diagram called "Arnold's tongues" delineating zones where the oscillator is stable from those where it is exponentially unstable (if the nonlinearity was absence). We can retrieve this diagram by calculating the steady states as a function of external detuning `)),t("mjx-container",Z,[(n(),e("svg",j,a[49]||(a[49]=[i('',1)]))),a[50]||(a[50]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"δ"),t("mo",null,"="),t("msub",null,[t("mi",null,"ω"),t("mi",null,"L")]),t("mo",null,"−"),t("msub",null,[t("mi",null,"ω"),t("mn",null,"0")])])],-1))]),a[54]||(a[54]=s(" and the parametric drive strength ")),t("mjx-container",B,[(n(),e("svg",A,a[51]||(a[51]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),a[52]||(a[52]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"λ")])],-1))]),a[55]||(a[55]=s("."))]),t("p",null,[a[60]||(a[60]=s("To perform a 2D sweep over driving frequency ")),t("mjx-container",q,[(n(),e("svg",J,a[56]||(a[56]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[57]||(a[57]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[61]||(a[61]=s(" and parametric drive strength ")),t("mjx-container",O,[(n(),e("svg",R,a[58]||(a[58]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),a[59]||(a[59]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"λ")])],-1))]),a[62]||(a[62]=s(", we keep ")),a[63]||(a[63]=t("code",null,"fixed",-1)),a[64]||(a[64]=s(" from before but include 2 variables in ")),a[65]||(a[65]=t("code",null,"varied",-1))]),a[74]||(a[74]=i(`
julia
varied = (ω => range(0.8, 1.2, 50), λ => range(0.001, 0.6, 50))
+result_2D = get_steady_states(harmonic_eq, varied, fixed);

+Solving for 2500 parameters...  53%|██████████▋         |  ETA: 0:00:01\x1B[K
+  # parameters solved:  1328\x1B[K
+  # paths tracked:      6640\x1B[K
+\x1B[A
+\x1B[A
+
+
+\x1B[K\x1B[A
+\x1B[K\x1B[A
+Solving for 2500 parameters...  83%|████████████████▋   |  ETA: 0:00:00\x1B[K
+  # parameters solved:  2085\x1B[K
+  # paths tracked:      10425\x1B[K
+\x1B[A
+\x1B[A
+
+
+\x1B[K\x1B[A
+\x1B[K\x1B[A
+Solving for 2500 parameters... 100%|████████████████████| Time: 0:00:01\x1B[K
+  # parameters solved:  2500\x1B[K
+  # paths tracked:      12500\x1B[K

Now, we count the number of solutions for each point and represent the corresponding phase diagram in parameter space. This is done using plot_phase_diagram. Only counting stable solutions,

julia
plot_phase_diagram(result_2D; class="stable")

In addition to phase diagrams, we can plot functions of the solution. The syntax is identical to 1D plotting. Let us overlay 2 branches into a single plot,

julia
# overlay branches with different colors
+plot(result_2D, "sqrt(u1^2 + v1^2)"; branch=1, class="stable", camera=(60, -40))
+plot!(result_2D, "sqrt(u1^2 + v1^2)"; branch=2, class="stable", color=:red)

Note that solutions are ordered in parameter space according to their closest neighbors. Plots can again be limited to a given class (e.g stable solutions only) through the keyword argument class.

This page was generated using Literate.jl.

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Three Wave Mixing vs four wave mixing ​

Packages ​

We load the following packages into our environment:

julia
using HarmonicBalance, Plots
+using Plots.Measures
+using Random
+
+Random.seed!(1234);
+Threads.nthreads() # check number of threads
1

system ​

julia
@variables β α ω ω0 F γ t x(t) # declare constant variables and a function x(t)
+diff_eq = DifferentialEquation(
+    d(x, t, 2) + ω0^2 * x + β * x^2 + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
+)
+
+add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)
+add_harmonic!(diff_eq, x, 2ω)
+add_harmonic!(diff_eq, x, 3ω)
+harmonic_eq = get_harmonic_equations(diff_eq)
A set of 6 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T), u3(T), v3(T)
+Parameters: ω, α, γ, ω0, β, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt) + u2(T)*cos(2ωt) + v2(T)*sin(2ωt) + u3(T)*cos(3ωt) + v3(T)*sin(3ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ + u1(T)*u2(T)*β - u1(T)*(ω^2) + u1(T)*(ω0^2) + v3(T)*v2(T)*β + v2(T)*v1(T)*β + v1(T)*γ*ω + u3(T)*u2(T)*β + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*u3(T)*α + (3//2)*u1(T)*(v3(T)^2)*α + (3//2)*u1(T)*v3(T)*v1(T)*α + (3//2)*u1(T)*(v2(T)^2)*α + (3//4)*u1(T)*(v1(T)^2)*α + (3//2)*u1(T)*(u3(T)^2)*α + (3//2)*u1(T)*(u2(T)^2)*α + (3//2)*v3(T)*v2(T)*u2(T)*α - (3//4)*(v2(T)^2)*u3(T)*α - (3//4)*(v1(T)^2)*u3(T)*α + (3//4)*u3(T)*(u2(T)^2)*α ~ F
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω + u1(T)*v2(T)*β - u1(T)*γ*ω + v3(T)*u2(T)*β - v2(T)*u3(T)*β - v1(T)*u2(T)*β - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v3(T)*α + (3//4)*(u1(T)^2)*v1(T)*α - (3//2)*u1(T)*v1(T)*u3(T)*α + (3//2)*(v3(T)^2)*v1(T)*α + (3//4)*v3(T)*(v2(T)^2)*α - (3//4)*v3(T)*(v1(T)^2)*α - (3//4)*v3(T)*(u2(T)^2)*α + (3//2)*(v2(T)^2)*v1(T)*α + (3//2)*v2(T)*u3(T)*u2(T)*α + (3//4)*(v1(T)^3)*α + (3//2)*v1(T)*(u3(T)^2)*α + (3//2)*v1(T)*(u2(T)^2)*α ~ 0
+
+Differential(T)(u2(T))*γ + (4//1)*Differential(T)(v2(T))*ω + (1//2)*(u1(T)^2)*β + u1(T)*u3(T)*β + v3(T)*v1(T)*β + (2//1)*v2(T)*γ*ω - (1//2)*(v1(T)^2)*β - (4//1)*u2(T)*(ω^2) + u2(T)*(ω0^2) + (3//2)*(u1(T)^2)*u2(T)*α + (3//2)*u1(T)*v3(T)*v2(T)*α + (3//2)*u1(T)*u3(T)*u2(T)*α + (3//2)*(v3(T)^2)*u2(T)*α - (3//2)*v3(T)*v1(T)*u2(T)*α + (3//4)*(v2(T)^2)*u2(T)*α + (3//2)*v2(T)*v1(T)*u3(T)*α + (3//2)*(v1(T)^2)*u2(T)*α + (3//2)*(u3(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0//1
+
+-(4//1)*Differential(T)(u2(T))*ω + Differential(T)(v2(T))*γ + u1(T)*v3(T)*β + u1(T)*v1(T)*β - (4//1)*v2(T)*(ω^2) + v2(T)*(ω0^2) - v1(T)*u3(T)*β - (2//1)*u2(T)*γ*ω + (3//2)*(u1(T)^2)*v2(T)*α + (3//2)*u1(T)*v3(T)*u2(T)*α - (3//2)*u1(T)*v2(T)*u3(T)*α + (3//2)*(v3(T)^2)*v2(T)*α + (3//2)*v3(T)*v2(T)*v1(T)*α + (3//4)*(v2(T)^3)*α + (3//2)*v2(T)*(v1(T)^2)*α + (3//2)*v2(T)*(u3(T)^2)*α + (3//4)*v2(T)*(u2(T)^2)*α + (3//2)*v1(T)*u3(T)*u2(T)*α ~ 0//1
+
+Differential(T)(u3(T))*γ + (6//1)*Differential(T)(v3(T))*ω + u1(T)*u2(T)*β + (3//1)*v3(T)*γ*ω - v2(T)*v1(T)*β - (9//1)*u3(T)*(ω^2) + u3(T)*(ω0^2) + (1//4)*(u1(T)^3)*α + (3//2)*(u1(T)^2)*u3(T)*α - (3//4)*u1(T)*(v2(T)^2)*α - (3//4)*u1(T)*(v1(T)^2)*α + (3//4)*u1(T)*(u2(T)^2)*α + (3//4)*(v3(T)^2)*u3(T)*α + (3//2)*(v2(T)^2)*u3(T)*α + (3//2)*v2(T)*v1(T)*u2(T)*α + (3//2)*(v1(T)^2)*u3(T)*α + (3//4)*(u3(T)^3)*α + (3//2)*u3(T)*(u2(T)^2)*α ~ 0//1
+
+-(6//1)*Differential(T)(u3(T))*ω + Differential(T)(v3(T))*γ + u1(T)*v2(T)*β - (9//1)*v3(T)*(ω^2) + v3(T)*(ω0^2) + v1(T)*u2(T)*β - (3//1)*u3(T)*γ*ω + (3//2)*(u1(T)^2)*v3(T)*α + (3//4)*(u1(T)^2)*v1(T)*α + (3//2)*u1(T)*v2(T)*u2(T)*α + (3//4)*(v3(T)^3)*α + (3//2)*v3(T)*(v2(T)^2)*α + (3//2)*v3(T)*(v1(T)^2)*α + (3//4)*v3(T)*(u3(T)^2)*α + (3//2)*v3(T)*(u2(T)^2)*α + (3//4)*(v2(T)^2)*v1(T)*α - (1//4)*(v1(T)^3)*α - (3//4)*v1(T)*(u2(T)^2)*α ~ 0//1

four wave mixing ​

`,9)),s("p",null,[i[4]||(i[4]=a("If we only have a cubic nonlineariy ")),s("mjx-container",o,[(e(),t("svg",d,i[0]||(i[0]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"α")])],-1))]),i[5]||(i[5]=a(", we observe the normal duffing oscillator response with no response at ")),s("mjx-container",T,[(e(),t("svg",Q,i[2]||(i[2]=[n('',1)]))),i[3]||(i[3]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"ω")])],-1))]),i[6]||(i[6]=a("."))]),i[54]||(i[54]=n(`
julia
varied = (ω => range(0.9, 1.2, 200)) # range of parameter values
+fixed = (α => 1.0, β => 0.0, ω0 => 1.0, γ => 0.005, F => 0.0025) # fixed parameters
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)# compute steady states
+
+p1 = plot(result; y="√(u1^2+v1^2)", legend=:best)
+p2 = plot(result; y="√(u2^2+v2^2)", legend=:best, ylims=(-0.1, 0.1))
+p3 = plot(result; y="√(u3^2+v3^2)", legend=:best)
+plot(p1, p2, p3; layout=(1, 3), size=(900, 300), margin=5mm)

Three wave mixing ​

',3)),s("p",null,[i[11]||(i[11]=a("If we only have a cubic nonlineariy ")),s("mjx-container",g,[(e(),t("svg",m,i[7]||(i[7]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),i[8]||(i[8]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"α")])],-1))]),i[12]||(i[12]=a(", we observe the normal duffing oscillator response with no response at ")),s("mjx-container",u,[(e(),t("svg",E,i[9]||(i[9]=[n('',1)]))),i[10]||(i[10]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"ω")])],-1))]),i[13]||(i[13]=a("."))]),s("p",null,[i[24]||(i[24]=a("We would like to investigate the three-wave mixing of the driven Duffing oscillator. This means we can excite the system resonantly if the oscillation frequencies ")),s("mjx-container",y,[(e(),t("svg",x,i[14]||(i[14]=[n('',1)]))),i[15]||(i[15]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")])])],-1))]),i[25]||(i[25]=a(" and ")),s("mjx-container",c,[(e(),t("svg",w,i[16]||(i[16]=[n('',1)]))),i[17]||(i[17]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")])])],-1))]),i[26]||(i[26]=a(" fullfil the conditions ")),s("mjx-container",v,[(e(),t("svg",F,i[18]||(i[18]=[n('',1)]))),i[19]||(i[19]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")]),s("mo",null,"±"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",null,"="),s("mo",null,"±"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")])])],-1))]),i[27]||(i[27]=a(". Here, we will especially focus on the degenerate three wave mixing, where ")),s("mjx-container",f,[(e(),t("svg",C,i[20]||(i[20]=[n('',1)]))),i[21]||(i[21]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")])])],-1))]),i[28]||(i[28]=a(" such that ")),s("mjx-container",b,[(e(),t("svg",D,i[22]||(i[22]=[n('',1)]))),i[23]||(i[23]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")]),s("mo",null,"="),s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")])])],-1))]),i[29]||(i[29]=a(". This is a very important process in quantum optics, since it allows us to generate photons with a frequency in the visible range from photons with a frequency in the infrared range. This is called frequency doubling and is used in many applications, e.g. in laser pointers."))]),i[55]||(i[55]=n(`
julia
varied = (ω => range(0.9, 1.2, 200))
+fixed = (α => 0.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025)
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+
+p1 = plot(result; y="√(u1^2+v1^2)", legend=:best)
+p2 = plot(result; y="√(u2^2+v2^2)", legend=:best, ylims=(-0.1, 0.1))
+p3 = plot(result; y="√(u3^2+v3^2)", legend=:best)
+plot(p1, p2, p3; layout=(1, 3), size=(900, 300), margin=5mm)

Both ​

',3)),s("p",null,[i[34]||(i[34]=a("If we only have a cubic nonlineariy ")),s("mjx-container",V,[(e(),t("svg",M,i[30]||(i[30]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),i[31]||(i[31]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"α")])],-1))]),i[35]||(i[35]=a(", we observe the normal duffing oscillator response with no response at ")),s("mjx-container",B,[(e(),t("svg",H,i[32]||(i[32]=[n('',1)]))),i[33]||(i[33]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"ω")])],-1))]),i[36]||(i[36]=a("."))]),s("p",null,[i[47]||(i[47]=a("We would like to investigate the three-wave mixing of the driven Duffing oscillator. This means we can excite the system resonantly if the oscillation frequencies ")),s("mjx-container",L,[(e(),t("svg",A,i[37]||(i[37]=[n('',1)]))),i[38]||(i[38]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")])])],-1))]),i[48]||(i[48]=a(" and ")),s("mjx-container",Z,[(e(),t("svg",j,i[39]||(i[39]=[n('',1)]))),i[40]||(i[40]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")])])],-1))]),i[49]||(i[49]=a(" fullfil the conditions ")),s("mjx-container",q,[(e(),t("svg",z,i[41]||(i[41]=[n('',1)]))),i[42]||(i[42]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")]),s("mo",null,"±"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",null,"="),s("mo",null,"±"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")])])],-1))]),i[50]||(i[50]=a(". Here, we will especially focus on the degenerate three wave mixing, where ")),s("mjx-container",S,[(e(),t("svg",J,i[43]||(i[43]=[n('',1)]))),i[44]||(i[44]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")])])],-1))]),i[51]||(i[51]=a(" such that ")),s("mjx-container",G,[(e(),t("svg",P,i[45]||(i[45]=[n('',1)]))),i[46]||(i[46]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")]),s("mo",null,"="),s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")])])],-1))]),i[52]||(i[52]=a(". This is a very important process in quantum optics, since it allows us to generate photons with a frequency in the visible range from photons with a frequency in the infrared range. This is called frequency doubling and is used in many applications, e.g. in laser pointers."))]),i[56]||(i[56]=n(`
julia
varied = (ω => range(0.9, 1.2, 200))
+fixed = (α => 1.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025)
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+
+p1 = plot(result; y="√(u1^2+v1^2)", legend=:best)
+p2 = plot(result; y="√(u2^2+v2^2)", legend=:best, ylims=(-0.1, 0.1))
+p3 = plot(result; y="√(u3^2+v3^2)", legend=:best)
+plot(p1, p2, p3; layout=(1, 3), size=(900, 300), margin=5mm)

This page was generated using Literate.jl.

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Three Wave Mixing vs four wave mixing ​

Packages ​

We load the following packages into our environment:

julia
using HarmonicBalance, Plots
+using Plots.Measures
+using Random
+
+Random.seed!(1234);
+Threads.nthreads() # check number of threads
1

system ​

julia
@variables β α ω ω0 F γ t x(t) # declare constant variables and a function x(t)
+diff_eq = DifferentialEquation(
+    d(x, t, 2) + ω0^2 * x + β * x^2 + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
+)
+
+add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)
+add_harmonic!(diff_eq, x, 2ω)
+add_harmonic!(diff_eq, x, 3ω)
+harmonic_eq = get_harmonic_equations(diff_eq)
A set of 6 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T), u3(T), v3(T)
+Parameters: ω, α, γ, ω0, β, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt) + u2(T)*cos(2ωt) + v2(T)*sin(2ωt) + u3(T)*cos(3ωt) + v3(T)*sin(3ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ + u1(T)*u2(T)*β - u1(T)*(ω^2) + u1(T)*(ω0^2) + v3(T)*v2(T)*β + v2(T)*v1(T)*β + v1(T)*γ*ω + u3(T)*u2(T)*β + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*u3(T)*α + (3//2)*u1(T)*(v3(T)^2)*α + (3//2)*u1(T)*v3(T)*v1(T)*α + (3//2)*u1(T)*(v2(T)^2)*α + (3//4)*u1(T)*(v1(T)^2)*α + (3//2)*u1(T)*(u3(T)^2)*α + (3//2)*u1(T)*(u2(T)^2)*α + (3//2)*v3(T)*v2(T)*u2(T)*α - (3//4)*(v2(T)^2)*u3(T)*α - (3//4)*(v1(T)^2)*u3(T)*α + (3//4)*u3(T)*(u2(T)^2)*α ~ F
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω + u1(T)*v2(T)*β - u1(T)*γ*ω + v3(T)*u2(T)*β - v2(T)*u3(T)*β - v1(T)*u2(T)*β - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v3(T)*α + (3//4)*(u1(T)^2)*v1(T)*α - (3//2)*u1(T)*v1(T)*u3(T)*α + (3//2)*(v3(T)^2)*v1(T)*α + (3//4)*v3(T)*(v2(T)^2)*α - (3//4)*v3(T)*(v1(T)^2)*α - (3//4)*v3(T)*(u2(T)^2)*α + (3//2)*(v2(T)^2)*v1(T)*α + (3//2)*v2(T)*u3(T)*u2(T)*α + (3//4)*(v1(T)^3)*α + (3//2)*v1(T)*(u3(T)^2)*α + (3//2)*v1(T)*(u2(T)^2)*α ~ 0
+
+Differential(T)(u2(T))*γ + (4//1)*Differential(T)(v2(T))*ω + (1//2)*(u1(T)^2)*β + u1(T)*u3(T)*β + v3(T)*v1(T)*β + (2//1)*v2(T)*γ*ω - (1//2)*(v1(T)^2)*β - (4//1)*u2(T)*(ω^2) + u2(T)*(ω0^2) + (3//2)*(u1(T)^2)*u2(T)*α + (3//2)*u1(T)*v3(T)*v2(T)*α + (3//2)*u1(T)*u3(T)*u2(T)*α + (3//2)*(v3(T)^2)*u2(T)*α - (3//2)*v3(T)*v1(T)*u2(T)*α + (3//4)*(v2(T)^2)*u2(T)*α + (3//2)*v2(T)*v1(T)*u3(T)*α + (3//2)*(v1(T)^2)*u2(T)*α + (3//2)*(u3(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0//1
+
+-(4//1)*Differential(T)(u2(T))*ω + Differential(T)(v2(T))*γ + u1(T)*v3(T)*β + u1(T)*v1(T)*β - (4//1)*v2(T)*(ω^2) + v2(T)*(ω0^2) - v1(T)*u3(T)*β - (2//1)*u2(T)*γ*ω + (3//2)*(u1(T)^2)*v2(T)*α + (3//2)*u1(T)*v3(T)*u2(T)*α - (3//2)*u1(T)*v2(T)*u3(T)*α + (3//2)*(v3(T)^2)*v2(T)*α + (3//2)*v3(T)*v2(T)*v1(T)*α + (3//4)*(v2(T)^3)*α + (3//2)*v2(T)*(v1(T)^2)*α + (3//2)*v2(T)*(u3(T)^2)*α + (3//4)*v2(T)*(u2(T)^2)*α + (3//2)*v1(T)*u3(T)*u2(T)*α ~ 0//1
+
+Differential(T)(u3(T))*γ + (6//1)*Differential(T)(v3(T))*ω + u1(T)*u2(T)*β + (3//1)*v3(T)*γ*ω - v2(T)*v1(T)*β - (9//1)*u3(T)*(ω^2) + u3(T)*(ω0^2) + (1//4)*(u1(T)^3)*α + (3//2)*(u1(T)^2)*u3(T)*α - (3//4)*u1(T)*(v2(T)^2)*α - (3//4)*u1(T)*(v1(T)^2)*α + (3//4)*u1(T)*(u2(T)^2)*α + (3//4)*(v3(T)^2)*u3(T)*α + (3//2)*(v2(T)^2)*u3(T)*α + (3//2)*v2(T)*v1(T)*u2(T)*α + (3//2)*(v1(T)^2)*u3(T)*α + (3//4)*(u3(T)^3)*α + (3//2)*u3(T)*(u2(T)^2)*α ~ 0//1
+
+-(6//1)*Differential(T)(u3(T))*ω + Differential(T)(v3(T))*γ + u1(T)*v2(T)*β - (9//1)*v3(T)*(ω^2) + v3(T)*(ω0^2) + v1(T)*u2(T)*β - (3//1)*u3(T)*γ*ω + (3//2)*(u1(T)^2)*v3(T)*α + (3//4)*(u1(T)^2)*v1(T)*α + (3//2)*u1(T)*v2(T)*u2(T)*α + (3//4)*(v3(T)^3)*α + (3//2)*v3(T)*(v2(T)^2)*α + (3//2)*v3(T)*(v1(T)^2)*α + (3//4)*v3(T)*(u3(T)^2)*α + (3//2)*v3(T)*(u2(T)^2)*α + (3//4)*(v2(T)^2)*v1(T)*α - (1//4)*(v1(T)^3)*α - (3//4)*v1(T)*(u2(T)^2)*α ~ 0//1

four wave mixing ​

`,9)),s("p",null,[i[4]||(i[4]=a("If we only have a cubic nonlineariy ")),s("mjx-container",o,[(e(),t("svg",d,i[0]||(i[0]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"α")])],-1))]),i[5]||(i[5]=a(", we observe the normal duffing oscillator response with no response at ")),s("mjx-container",T,[(e(),t("svg",Q,i[2]||(i[2]=[n('',1)]))),i[3]||(i[3]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"ω")])],-1))]),i[6]||(i[6]=a("."))]),i[54]||(i[54]=n(`
julia
varied = (ω => range(0.9, 1.2, 200)) # range of parameter values
+fixed = (α => 1.0, β => 0.0, ω0 => 1.0, γ => 0.005, F => 0.0025) # fixed parameters
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)# compute steady states
+
+p1 = plot(result; y="√(u1^2+v1^2)", legend=:best)
+p2 = plot(result; y="√(u2^2+v2^2)", legend=:best, ylims=(-0.1, 0.1))
+p3 = plot(result; y="√(u3^2+v3^2)", legend=:best)
+plot(p1, p2, p3; layout=(1, 3), size=(900, 300), margin=5mm)

Three wave mixing ​

',3)),s("p",null,[i[11]||(i[11]=a("If we only have a cubic nonlineariy ")),s("mjx-container",g,[(e(),t("svg",m,i[7]||(i[7]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),i[8]||(i[8]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"α")])],-1))]),i[12]||(i[12]=a(", we observe the normal duffing oscillator response with no response at ")),s("mjx-container",u,[(e(),t("svg",E,i[9]||(i[9]=[n('',1)]))),i[10]||(i[10]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"ω")])],-1))]),i[13]||(i[13]=a("."))]),s("p",null,[i[24]||(i[24]=a("We would like to investigate the three-wave mixing of the driven Duffing oscillator. This means we can excite the system resonantly if the oscillation frequencies ")),s("mjx-container",y,[(e(),t("svg",x,i[14]||(i[14]=[n('',1)]))),i[15]||(i[15]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")])])],-1))]),i[25]||(i[25]=a(" and ")),s("mjx-container",c,[(e(),t("svg",w,i[16]||(i[16]=[n('',1)]))),i[17]||(i[17]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")])])],-1))]),i[26]||(i[26]=a(" fullfil the conditions ")),s("mjx-container",v,[(e(),t("svg",F,i[18]||(i[18]=[n('',1)]))),i[19]||(i[19]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")]),s("mo",null,"±"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",null,"="),s("mo",null,"±"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")])])],-1))]),i[27]||(i[27]=a(". Here, we will especially focus on the degenerate three wave mixing, where ")),s("mjx-container",f,[(e(),t("svg",C,i[20]||(i[20]=[n('',1)]))),i[21]||(i[21]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")])])],-1))]),i[28]||(i[28]=a(" such that ")),s("mjx-container",b,[(e(),t("svg",D,i[22]||(i[22]=[n('',1)]))),i[23]||(i[23]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")]),s("mo",null,"="),s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")])])],-1))]),i[29]||(i[29]=a(". This is a very important process in quantum optics, since it allows us to generate photons with a frequency in the visible range from photons with a frequency in the infrared range. This is called frequency doubling and is used in many applications, e.g. in laser pointers."))]),i[55]||(i[55]=n(`
julia
varied = (ω => range(0.9, 1.2, 200))
+fixed = (α => 0.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025)
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+
+p1 = plot(result; y="√(u1^2+v1^2)", legend=:best)
+p2 = plot(result; y="√(u2^2+v2^2)", legend=:best, ylims=(-0.1, 0.1))
+p3 = plot(result; y="√(u3^2+v3^2)", legend=:best)
+plot(p1, p2, p3; layout=(1, 3), size=(900, 300), margin=5mm)

Both ​

',3)),s("p",null,[i[34]||(i[34]=a("If we only have a cubic nonlineariy ")),s("mjx-container",V,[(e(),t("svg",M,i[30]||(i[30]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),i[31]||(i[31]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"α")])],-1))]),i[35]||(i[35]=a(", we observe the normal duffing oscillator response with no response at ")),s("mjx-container",B,[(e(),t("svg",H,i[32]||(i[32]=[n('',1)]))),i[33]||(i[33]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("mi",null,"ω")])],-1))]),i[36]||(i[36]=a("."))]),s("p",null,[i[47]||(i[47]=a("We would like to investigate the three-wave mixing of the driven Duffing oscillator. This means we can excite the system resonantly if the oscillation frequencies ")),s("mjx-container",L,[(e(),t("svg",A,i[37]||(i[37]=[n('',1)]))),i[38]||(i[38]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")])])],-1))]),i[48]||(i[48]=a(" and ")),s("mjx-container",Z,[(e(),t("svg",j,i[39]||(i[39]=[n('',1)]))),i[40]||(i[40]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")])])],-1))]),i[49]||(i[49]=a(" fullfil the conditions ")),s("mjx-container",q,[(e(),t("svg",z,i[41]||(i[41]=[n('',1)]))),i[42]||(i[42]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")]),s("mo",null,"±"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",null,"="),s("mo",null,"±"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")])])],-1))]),i[50]||(i[50]=a(". Here, we will especially focus on the degenerate three wave mixing, where ")),s("mjx-container",S,[(e(),t("svg",J,i[43]||(i[43]=[n('',1)]))),i[44]||(i[44]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"ω"),s("mn",null,"2")]),s("mo",null,"="),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")])])],-1))]),i[51]||(i[51]=a(" such that ")),s("mjx-container",G,[(e(),t("svg",P,i[45]||(i[45]=[n('',1)]))),i[46]||(i[46]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"2"),s("msub",null,[s("mi",null,"ω"),s("mn",null,"0")]),s("mo",null,"="),s("msub",null,[s("mi",null,"ω"),s("mn",null,"1")])])],-1))]),i[52]||(i[52]=a(". This is a very important process in quantum optics, since it allows us to generate photons with a frequency in the visible range from photons with a frequency in the infrared range. This is called frequency doubling and is used in many applications, e.g. in laser pointers."))]),i[56]||(i[56]=n(`
julia
varied = (ω => range(0.9, 1.2, 200))
+fixed = (α => 1.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025)
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+
+p1 = plot(result; y="√(u1^2+v1^2)", legend=:best)
+p2 = plot(result; y="√(u2^2+v2^2)", legend=:best, ylims=(-0.1, 0.1))
+p3 = plot(result; y="√(u3^2+v3^2)", legend=:best)
+plot(p1, p2, p3; layout=(1, 3), size=(900, 300), margin=5mm)

This page was generated using Literate.jl.

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If you use HarmonicBalance.jl in your project, we kindly ask you to cite this paper, namely:

HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance, Jan Košata, Javier del Pino, Toni L. Heugel, Oded Zilberberg, SciPost Phys. Codebases 6 (2022)

The limit cycle finding algorithm is based on the work of this paper:

Limit cycles as stationary states of an extended harmonic balance ansatz J. del Pino, J. Košata, and O. Zilberberg, Phys. Rev. Res. 6, 033180 (2024).

',5)]))}const m=e(n,[["render",r]]);export{u as __pageData,m as default}; diff --git a/v0.10.7/assets/introduction_citation.md.iwhtmYXt.lean.js b/v0.10.7/assets/introduction_citation.md.iwhtmYXt.lean.js new file mode 100644 index 00000000..d4a8fde6 --- /dev/null +++ b/v0.10.7/assets/introduction_citation.md.iwhtmYXt.lean.js @@ -0,0 +1 @@ +import{_ as e,c as a,a4 as o,o as i}from"./chunks/framework.D3a-l5sU.js";const u=JSON.parse('{"title":"Citation","description":"","frontmatter":{},"headers":[],"relativePath":"introduction/citation.md","filePath":"introduction/citation.md"}'),n={name:"introduction/citation.md"};function r(s,t,c,l,d,p){return i(),a("div",null,t[0]||(t[0]=[o('

Citation ​

If you use HarmonicBalance.jl in your project, we kindly ask you to cite this paper, namely:

HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance, Jan Košata, Javier del Pino, Toni L. Heugel, Oded Zilberberg, SciPost Phys. Codebases 6 (2022)

The limit cycle finding algorithm is based on the work of this paper:

Limit cycles as stationary states of an extended harmonic balance ansatz J. del Pino, J. Košata, and O. Zilberberg, Phys. Rev. Res. 6, 033180 (2024).

',5)]))}const m=e(n,[["render",r]]);export{u as __pageData,m as default}; diff --git a/v0.10.7/assets/introduction_index.md.B8BijThY.js b/v0.10.7/assets/introduction_index.md.B8BijThY.js new file mode 100644 index 00000000..917fd9aa --- /dev/null +++ b/v0.10.7/assets/introduction_index.md.B8BijThY.js @@ -0,0 +1,21 @@ +import{_ as i,c as s,a4 as Q,j as a,o as T}from"./chunks/framework.D3a-l5sU.js";const e="/HarmonicBalance.jl/v0.10.7/assets/oxznusu.B1eISI2b.png",V=JSON.parse('{"title":"Installation","description":"","frontmatter":{},"headers":[],"relativePath":"introduction/index.md","filePath":"introduction/index.md"}'),n={name:"introduction/index.md"},l={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},h={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-4.03ex"},xmlns:"http://www.w3.org/2000/svg",width:"48.629ex",height:"6.03ex",role:"img",focusable:"false",viewBox:"0 -883.9 21494.2 2665.1","aria-hidden":"true"};function d(r,t,p,o,k,m){return T(),s("div",null,[t[2]||(t[2]=Q(`

Installation ​

It is easy to install HarmonicBalance.jl as we are registered in the Julia General registry. You can simply run the following command in the Julia REPL:

julia
julia> using Pkg
+julia> Pkg.add("HarmonicBalance")

or

julia
julia> ] # \`]\` should be pressed
+julia> Pkg.add("HarmonicBalance")

You can check which version you have installled with the command

julia
julia> ]
+julia> status HarmonicBalance

Getting Started ​

Let us find the steady states of an external driven Duffing oscillator with nonlinear damping. Its equation of motion is:

`,9)),a("mjx-container",l,[(T(),s("svg",h,t[0]||(t[0]=[Q('',1)]))),t[1]||(t[1]=a("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[a("munder",null,[a("mrow",{"data-mjx-texclass":"OP"},[a("munder",null,[a("mrow",null,[a("mrow",{"data-mjx-texclass":"ORD"},[a("mover",null,[a("mi",null,"x"),a("mo",null,"¨")])]),a("mo",{stretchy:"false"},"("),a("mi",null,"t"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mi",null,"γ"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mover",null,[a("mi",null,"x"),a("mo",null,"˙")])]),a("mo",{stretchy:"false"},"("),a("mi",null,"t"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("msubsup",null,[a("mi",null,"ω"),a("mn",null,"0"),a("mn",null,"2")]),a("mi",null,"x"),a("mo",{stretchy:"false"},"("),a("mi",null,"t"),a("mo",{stretchy:"false"},")")]),a("mo",null,"⏟")])]),a("mrow",{"data-mjx-texclass":"ORD"},[a("mtext",null,"damped harmonic oscillator")])]),a("mo",null,"+"),a("munder",null,[a("mrow",{"data-mjx-texclass":"OP"},[a("munder",null,[a("mrow",null,[a("mi",null,"α"),a("mi",null,"x"),a("mo",{stretchy:"false"},"("),a("mi",null,"t"),a("msup",null,[a("mo",{stretchy:"false"},")"),a("mn",null,"3")])]),a("mo",null,"⏟")])]),a("mrow",{"data-mjx-texclass":"ORD"},[a("mtext",null,"Duffing coefficient")])]),a("mo",null,"="),a("munder",null,[a("mrow",{"data-mjx-texclass":"OP"},[a("munder",null,[a("mrow",null,[a("mi",null,"F"),a("mi",null,"cos"),a("mo",{"data-mjx-texclass":"NONE"},"⁡"),a("mo",{stretchy:"false"},"("),a("mi",null,"ω"),a("mi",null,"t"),a("mo",{stretchy:"false"},")")]),a("mo",null,"⏟")])]),a("mrow",{"data-mjx-texclass":"ORD"},[a("mtext",null,"periodic drive")])])])],-1))]),t[3]||(t[3]=Q(`
julia
using HarmonicBalance
+@variables α ω ω0 F t η x(t) # declare constant variables and a function x(t)
+eom = d(x,t,2) + ω0^2*x + α*x^3 + η*d(x,t)*x^2 ~ F*cos(ω*t)
+diff_eq = DifferentialEquation(eom, x)
+add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)
+
+# implement ansatz to get harmonic equations
+harmonic_eq = get_harmonic_equations(diff_eq)
+
+fixed = (α => 1.0, ω0 => 1.0, F => 0.01, η => 0.1)   # fixed parameters
+varied = ω => range(0.9, 1.2, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
A steady state result for 100 parameter points
+
+Solution branches:   3
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels

The obtained steady states can be plotted as a function of the driving frequency:

julia
plot(result, "sqrt(u1^2 + v1^2)")

If you want learn more on what you can do with HarmonicBalance.jl, check out the tutorials. We also have collected some examples of different physical systems.

',6))])}const g=i(n,[["render",d]]);export{V as __pageData,g as default}; diff --git a/v0.10.7/assets/introduction_index.md.B8BijThY.lean.js b/v0.10.7/assets/introduction_index.md.B8BijThY.lean.js new file mode 100644 index 00000000..917fd9aa --- /dev/null +++ b/v0.10.7/assets/introduction_index.md.B8BijThY.lean.js @@ -0,0 +1,21 @@ +import{_ as i,c as s,a4 as Q,j as a,o as T}from"./chunks/framework.D3a-l5sU.js";const e="/HarmonicBalance.jl/v0.10.7/assets/oxznusu.B1eISI2b.png",V=JSON.parse('{"title":"Installation","description":"","frontmatter":{},"headers":[],"relativePath":"introduction/index.md","filePath":"introduction/index.md"}'),n={name:"introduction/index.md"},l={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},h={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-4.03ex"},xmlns:"http://www.w3.org/2000/svg",width:"48.629ex",height:"6.03ex",role:"img",focusable:"false",viewBox:"0 -883.9 21494.2 2665.1","aria-hidden":"true"};function d(r,t,p,o,k,m){return T(),s("div",null,[t[2]||(t[2]=Q(`

Installation ​

It is easy to install HarmonicBalance.jl as we are registered in the Julia General registry. You can simply run the following command in the Julia REPL:

julia
julia> using Pkg
+julia> Pkg.add("HarmonicBalance")

or

julia
julia> ] # \`]\` should be pressed
+julia> Pkg.add("HarmonicBalance")

You can check which version you have installled with the command

julia
julia> ]
+julia> status HarmonicBalance

Getting Started ​

Let us find the steady states of an external driven Duffing oscillator with nonlinear damping. Its equation of motion is:

`,9)),a("mjx-container",l,[(T(),s("svg",h,t[0]||(t[0]=[Q('',1)]))),t[1]||(t[1]=a("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[a("munder",null,[a("mrow",{"data-mjx-texclass":"OP"},[a("munder",null,[a("mrow",null,[a("mrow",{"data-mjx-texclass":"ORD"},[a("mover",null,[a("mi",null,"x"),a("mo",null,"¨")])]),a("mo",{stretchy:"false"},"("),a("mi",null,"t"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("mi",null,"γ"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mover",null,[a("mi",null,"x"),a("mo",null,"˙")])]),a("mo",{stretchy:"false"},"("),a("mi",null,"t"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("msubsup",null,[a("mi",null,"ω"),a("mn",null,"0"),a("mn",null,"2")]),a("mi",null,"x"),a("mo",{stretchy:"false"},"("),a("mi",null,"t"),a("mo",{stretchy:"false"},")")]),a("mo",null,"⏟")])]),a("mrow",{"data-mjx-texclass":"ORD"},[a("mtext",null,"damped harmonic oscillator")])]),a("mo",null,"+"),a("munder",null,[a("mrow",{"data-mjx-texclass":"OP"},[a("munder",null,[a("mrow",null,[a("mi",null,"α"),a("mi",null,"x"),a("mo",{stretchy:"false"},"("),a("mi",null,"t"),a("msup",null,[a("mo",{stretchy:"false"},")"),a("mn",null,"3")])]),a("mo",null,"⏟")])]),a("mrow",{"data-mjx-texclass":"ORD"},[a("mtext",null,"Duffing coefficient")])]),a("mo",null,"="),a("munder",null,[a("mrow",{"data-mjx-texclass":"OP"},[a("munder",null,[a("mrow",null,[a("mi",null,"F"),a("mi",null,"cos"),a("mo",{"data-mjx-texclass":"NONE"},"⁡"),a("mo",{stretchy:"false"},"("),a("mi",null,"ω"),a("mi",null,"t"),a("mo",{stretchy:"false"},")")]),a("mo",null,"⏟")])]),a("mrow",{"data-mjx-texclass":"ORD"},[a("mtext",null,"periodic drive")])])])],-1))]),t[3]||(t[3]=Q(`
julia
using HarmonicBalance
+@variables α ω ω0 F t η x(t) # declare constant variables and a function x(t)
+eom = d(x,t,2) + ω0^2*x + α*x^3 + η*d(x,t)*x^2 ~ F*cos(ω*t)
+diff_eq = DifferentialEquation(eom, x)
+add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)
+
+# implement ansatz to get harmonic equations
+harmonic_eq = get_harmonic_equations(diff_eq)
+
+fixed = (α => 1.0, ω0 => 1.0, F => 0.01, η => 0.1)   # fixed parameters
+varied = ω => range(0.9, 1.2, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
A steady state result for 100 parameter points
+
+Solution branches:   3
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels

The obtained steady states can be plotted as a function of the driving frequency:

julia
plot(result, "sqrt(u1^2 + v1^2)")

If you want learn more on what you can do with HarmonicBalance.jl, check out the tutorials. We also have collected some examples of different physical systems.

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a/v0.10.7/assets/manual_Krylov-Bogoliubov_method.md.D_dGJgze.js b/v0.10.7/assets/manual_Krylov-Bogoliubov_method.md.D_dGJgze.js new file mode 100644 index 00000000..6e26a45e --- /dev/null +++ b/v0.10.7/assets/manual_Krylov-Bogoliubov_method.md.D_dGJgze.js @@ -0,0 +1,29 @@ +import{_ as h,c as n,j as s,a,a4 as t,o as e}from"./chunks/framework.D3a-l5sU.js";const F=JSON.parse('{"title":"Krylov-Bogoliubov Averaging Method","description":"","frontmatter":{},"headers":[],"relativePath":"manual/Krylov-Bogoliubov_method.md","filePath":"manual/Krylov-Bogoliubov_method.md"}'),l={name:"manual/Krylov-Bogoliubov_method.md"},k={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.67ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 1622 1000","aria-hidden":"true"};function r(o,i,d,g,E,y){return e(),n("div",null,[i[8]||(i[8]=s("h1",{id:"Krylov-Bogoliubov",tabindex:"-1"},[a("Krylov-Bogoliubov Averaging Method "),s("a",{class:"header-anchor",href:"#Krylov-Bogoliubov","aria-label":'Permalink to "Krylov-Bogoliubov Averaging Method {#Krylov-Bogoliubov}"'},"​")],-1)),s("p",null,[i[2]||(i[2]=a("The Krylov-Bogoliubov averaging method is an alternative high-frequency expansion technique used to analyze dynamical systems. Unlike the ")),i[3]||(i[3]=s("a",{href:"https://en.wikipedia.org/wiki/Harmonic_balance",target:"_blank",rel:"noreferrer"},"Harmonic Balance method",-1)),i[4]||(i[4]=a(", which is detailed in the ")),i[5]||(i[5]=s("a",{href:"/HarmonicBalance.jl/v0.10.7/background/harmonic_balance#intro_hb"},"background section",-1)),i[6]||(i[6]=a(", the Krylov-Bogoliubov method excels in computing higher orders in ")),s("mjx-container",k,[(e(),n("svg",p,i[0]||(i[0]=[t('',1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"1"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"ω")])],-1))]),i[7]||(i[7]=a(", enabling the capture of faster dynamics within a system."))]),i[9]||(i[9]=t(`

    The primary advantage of the Krylov-Bogoliubov method lies in its ability to delve deeper into high-frequency components, allowing a more comprehensive understanding of fast dynamical behaviors. By leveraging this technique, one can obtain higher-order approximations that shed light on intricate system dynamics.

    However, it's essential to note a limitation: this method cannot handle multiple harmonics within a single variable, unlike some other high-frequency expansion methods.

    Usage ​

    To compute the Krylov-Bogoliubov averaging method within your system, utilize the function get_krylov_equations. This function is designed specifically to implement the methodology and derive the equations necessary to analyze the system dynamics using this technique.

    Function Reference ​

    # HarmonicBalance.KrylovBogoliubov.get_krylov_equations — Function.
    julia
    get_krylov_equations(
+    diff_eom::DifferentialEquation;
+    order,
+    fast_time,
+    slow_time
+)

    Apply the Krylov-Bogoliubov averaging method to a specific order to obtain a set of ODEs (the slow-flow equations) governing the harmonics of diff_eom.

    The harmonics evolve in slow_time, the oscillating terms themselves in fast_time. If no input is used, a variable T is defined for slow_time and fast_time is taken as the independent variable of diff_eom.

    Krylov-Bogoliubov averaging method can be applied up to order = 2.

    Example

    julia
    julia> @variables t, x(t), ω0, ω, F;
+
+# enter the simple harmonic oscillator
+julia> diff_eom = DifferentialEquation( d(x,t,2) + ω0^2 * x ~ F *cos(ω*t), x);
+
+# expand x in the harmonic ω
+julia> add_harmonic!(diff_eom, x, ω);
+
+# get equations for the harmonics evolving in the slow time T to first order
+julia> harmonic_eom = get_krylov_equations(diff_eom, order = 1)
+
+A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, F, ω0
+
+Harmonic ansatz:
+xˍt(t) =
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+((1//2)*(ω^2)*v1(T) - (1//2)*(ω0^2)*v1(T)) / ω ~ Differential(T)(u1(T))
+
+((1//2)*(ω0^2)*u1(T) - (1//2)*F - (1//2)*(ω^2)*u1(T)) / ω ~ Differential(T)(v1(T))

    source


    For further information and a detailed understanding of this method, refer to Krylov-Bogoliubov averaging method on Wikipedia.

    `,9))])}const m=h(l,[["render",r]]);export{F as __pageData,m as default}; diff --git a/v0.10.7/assets/manual_Krylov-Bogoliubov_method.md.D_dGJgze.lean.js b/v0.10.7/assets/manual_Krylov-Bogoliubov_method.md.D_dGJgze.lean.js new file mode 100644 index 00000000..6e26a45e --- /dev/null +++ b/v0.10.7/assets/manual_Krylov-Bogoliubov_method.md.D_dGJgze.lean.js @@ -0,0 +1,29 @@ +import{_ as h,c as n,j as s,a,a4 as t,o as e}from"./chunks/framework.D3a-l5sU.js";const F=JSON.parse('{"title":"Krylov-Bogoliubov Averaging Method","description":"","frontmatter":{},"headers":[],"relativePath":"manual/Krylov-Bogoliubov_method.md","filePath":"manual/Krylov-Bogoliubov_method.md"}'),l={name:"manual/Krylov-Bogoliubov_method.md"},k={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.67ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 1622 1000","aria-hidden":"true"};function r(o,i,d,g,E,y){return e(),n("div",null,[i[8]||(i[8]=s("h1",{id:"Krylov-Bogoliubov",tabindex:"-1"},[a("Krylov-Bogoliubov Averaging Method "),s("a",{class:"header-anchor",href:"#Krylov-Bogoliubov","aria-label":'Permalink to "Krylov-Bogoliubov Averaging Method {#Krylov-Bogoliubov}"'},"​")],-1)),s("p",null,[i[2]||(i[2]=a("The Krylov-Bogoliubov averaging method is an alternative high-frequency expansion technique used to analyze dynamical systems. Unlike the ")),i[3]||(i[3]=s("a",{href:"https://en.wikipedia.org/wiki/Harmonic_balance",target:"_blank",rel:"noreferrer"},"Harmonic Balance method",-1)),i[4]||(i[4]=a(", which is detailed in the ")),i[5]||(i[5]=s("a",{href:"/HarmonicBalance.jl/v0.10.7/background/harmonic_balance#intro_hb"},"background section",-1)),i[6]||(i[6]=a(", the Krylov-Bogoliubov method excels in computing higher orders in ")),s("mjx-container",k,[(e(),n("svg",p,i[0]||(i[0]=[t('',1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mn",null,"1"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"ω")])],-1))]),i[7]||(i[7]=a(", enabling the capture of faster dynamics within a system."))]),i[9]||(i[9]=t(`

    Purpose and Advantages ​

    The primary advantage of the Krylov-Bogoliubov method lies in its ability to delve deeper into high-frequency components, allowing a more comprehensive understanding of fast dynamical behaviors. By leveraging this technique, one can obtain higher-order approximations that shed light on intricate system dynamics.

    However, it's essential to note a limitation: this method cannot handle multiple harmonics within a single variable, unlike some other high-frequency expansion methods.

    Usage ​

    To compute the Krylov-Bogoliubov averaging method within your system, utilize the function get_krylov_equations. This function is designed specifically to implement the methodology and derive the equations necessary to analyze the system dynamics using this technique.

    Function Reference ​

    # HarmonicBalance.KrylovBogoliubov.get_krylov_equations — Function.
    julia
    get_krylov_equations(
+    diff_eom::DifferentialEquation;
+    order,
+    fast_time,
+    slow_time
+)

    Apply the Krylov-Bogoliubov averaging method to a specific order to obtain a set of ODEs (the slow-flow equations) governing the harmonics of diff_eom.

    The harmonics evolve in slow_time, the oscillating terms themselves in fast_time. If no input is used, a variable T is defined for slow_time and fast_time is taken as the independent variable of diff_eom.

    Krylov-Bogoliubov averaging method can be applied up to order = 2.

    Example

    julia
    julia> @variables t, x(t), ω0, ω, F;
+
+# enter the simple harmonic oscillator
+julia> diff_eom = DifferentialEquation( d(x,t,2) + ω0^2 * x ~ F *cos(ω*t), x);
+
+# expand x in the harmonic ω
+julia> add_harmonic!(diff_eom, x, ω);
+
+# get equations for the harmonics evolving in the slow time T to first order
+julia> harmonic_eom = get_krylov_equations(diff_eom, order = 1)
+
+A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, F, ω0
+
+Harmonic ansatz:
+xˍt(t) =
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+((1//2)*(ω^2)*v1(T) - (1//2)*(ω0^2)*v1(T)) / ω ~ Differential(T)(u1(T))
+
+((1//2)*(ω0^2)*u1(T) - (1//2)*F - (1//2)*(ω^2)*u1(T)) / ω ~ Differential(T)(v1(T))

    source


    For further information and a detailed understanding of this method, refer to Krylov-Bogoliubov averaging method on Wikipedia.

    `,9))])}const m=h(l,[["render",r]]);export{F as __pageData,m as default}; diff --git a/v0.10.7/assets/manual_entering_eom.md.BqKYmtjt.js b/v0.10.7/assets/manual_entering_eom.md.BqKYmtjt.js new file mode 100644 index 00000000..9508e7a6 --- /dev/null +++ b/v0.10.7/assets/manual_entering_eom.md.BqKYmtjt.js @@ -0,0 +1,18 @@ +import{_ as s,c as a,a4 as n,o as t}from"./chunks/framework.D3a-l5sU.js";const o=JSON.parse('{"title":"Entering equations of motion","description":"","frontmatter":{},"headers":[],"relativePath":"manual/entering_eom.md","filePath":"manual/entering_eom.md"}'),e={name:"manual/entering_eom.md"};function h(l,i,k,p,r,d){return t(),a("div",null,i[0]||(i[0]=[n(`

    Entering equations of motion ​

    The struct DifferentialEquation is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using add_harmonic!.

    DifferentialEquation.equations stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.

    # HarmonicBalance.DifferentialEquation — Type.
    julia
    mutable struct DifferentialEquation

    Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for HarmonicBalance.jl ; after inputting the equations, the harmonics ansatz needs to be specified using add_harmonic!.

    Fields

    • equations::OrderedCollections.OrderedDict{Num, Equation}: Assigns to each variable an equation of motion.

    • harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}: Assigns to each variable a set of harmonics.

    Example

    julia
    julia> @variables t, x(t), y(t), ω0, ω, F, k;
+
+# equivalent ways to enter the simple harmonic oscillator
+julia> DifferentialEquation(d(x,t,2) + ω0^2 * x - F * cos(ω*t), x);
+julia> DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x);
+
+# two coupled oscillators, one of them driven
+julia> DifferentialEquation([d(x,t,2) + ω0^2 * x - k*y, d(y,t,2) + ω0^2 * y - k*x] .~ [F * cos(ω*t), 0], [x,y]);

    source


    # HarmonicBalance.add_harmonic! — Function.
    julia
    add_harmonic!(diff_eom::DifferentialEquation, var::Num, ω)

    Add the harmonic ω to the harmonic ansatz used to expand the variable var in diff_eom.

    Example

    define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω

    julia
    julia> @variables t, x(t), y(t), ω0, ω, F, k;
+julia> diff_eq = DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x);
+julia> add_harmonic!(diff_eq, x, ω) # expand x using ω
+
+System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ω;
+
+(ω0^2)*x(t) + Differential(t)(Differential(t)(x(t))) ~ F*cos(t*ω)

    source


    # Symbolics.get_variables — Method.
    julia
    get_variables(diff_eom::DifferentialEquation) -> Vector{Num}

    Return the dependent variables of diff_eom.

    source


    # HarmonicBalance.get_independent_variables — Method.
    julia
    get_independent_variables(
+    diff_eom::DifferentialEquation
+) -> Any

    Return the independent dependent variables of diff_eom.

    source


    `,11)]))}const g=s(e,[["render",h]]);export{o as __pageData,g as default}; diff --git a/v0.10.7/assets/manual_entering_eom.md.BqKYmtjt.lean.js b/v0.10.7/assets/manual_entering_eom.md.BqKYmtjt.lean.js new file mode 100644 index 00000000..9508e7a6 --- /dev/null +++ b/v0.10.7/assets/manual_entering_eom.md.BqKYmtjt.lean.js @@ -0,0 +1,18 @@ +import{_ as s,c as a,a4 as n,o as t}from"./chunks/framework.D3a-l5sU.js";const o=JSON.parse('{"title":"Entering equations of motion","description":"","frontmatter":{},"headers":[],"relativePath":"manual/entering_eom.md","filePath":"manual/entering_eom.md"}'),e={name:"manual/entering_eom.md"};function h(l,i,k,p,r,d){return t(),a("div",null,i[0]||(i[0]=[n(`

    Entering equations of motion ​

    The struct DifferentialEquation is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using add_harmonic!.

    DifferentialEquation.equations stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.

    # HarmonicBalance.DifferentialEquation — Type.
    julia
    mutable struct DifferentialEquation

    Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for HarmonicBalance.jl ; after inputting the equations, the harmonics ansatz needs to be specified using add_harmonic!.

    Fields

    • equations::OrderedCollections.OrderedDict{Num, Equation}: Assigns to each variable an equation of motion.

    • harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}: Assigns to each variable a set of harmonics.

    Example

    julia
    julia> @variables t, x(t), y(t), ω0, ω, F, k;
+
+# equivalent ways to enter the simple harmonic oscillator
+julia> DifferentialEquation(d(x,t,2) + ω0^2 * x - F * cos(ω*t), x);
+julia> DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x);
+
+# two coupled oscillators, one of them driven
+julia> DifferentialEquation([d(x,t,2) + ω0^2 * x - k*y, d(y,t,2) + ω0^2 * y - k*x] .~ [F * cos(ω*t), 0], [x,y]);

    source


    # HarmonicBalance.add_harmonic! — Function.
    julia
    add_harmonic!(diff_eom::DifferentialEquation, var::Num, ω)

    Add the harmonic ω to the harmonic ansatz used to expand the variable var in diff_eom.

    Example

    define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω

    julia
    julia> @variables t, x(t), y(t), ω0, ω, F, k;
+julia> diff_eq = DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x);
+julia> add_harmonic!(diff_eq, x, ω) # expand x using ω
+
+System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ω;
+
+(ω0^2)*x(t) + Differential(t)(Differential(t)(x(t))) ~ F*cos(t*ω)

    source


    # Symbolics.get_variables — Method.
    julia
    get_variables(diff_eom::DifferentialEquation) -> Vector{Num}

    Return the dependent variables of diff_eom.

    source


    # HarmonicBalance.get_independent_variables — Method.
    julia
    get_independent_variables(
+    diff_eom::DifferentialEquation
+) -> Any

    Return the independent dependent variables of diff_eom.

    source


    `,11)]))}const g=s(e,[["render",h]]);export{o as __pageData,g as default}; diff --git a/v0.10.7/assets/manual_extracting_harmonics.md.euttv_oy.js b/v0.10.7/assets/manual_extracting_harmonics.md.euttv_oy.js new file mode 100644 index 00000000..38f19ce5 --- /dev/null +++ b/v0.10.7/assets/manual_extracting_harmonics.md.euttv_oy.js @@ -0,0 +1,32 @@ +import{_ as l,c as t,a4 as e,j as a,a as i,o as n}from"./chunks/framework.D3a-l5sU.js";const Z=JSON.parse('{"title":"Extracting harmonic equations","description":"","frontmatter":{},"headers":[],"relativePath":"manual/extracting_harmonics.md","filePath":"manual/extracting_harmonics.md"}'),r={name:"manual/extracting_harmonics.md"},o={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},Q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.972ex"},xmlns:"http://www.w3.org/2000/svg",width:"47.051ex",height:"3.144ex",role:"img",focusable:"false",viewBox:"0 -960 20796.4 1389.6","aria-hidden":"true"},T={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},h={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.666ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.845ex",height:"2.363ex",role:"img",focusable:"false",viewBox:"0 -750 3909.4 1044.2","aria-hidden":"true"},d={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.666ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.251ex",height:"1.668ex",role:"img",focusable:"false",viewBox:"0 -443 1436.9 737.2","aria-hidden":"true"},k={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.611ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2038 1000","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.611ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2038 1000","aria-hidden":"true"},u={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.378ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 1051 683","aria-hidden":"true"},E={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},x={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.009ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 888 683","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},f={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"5.518ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 2439 683","aria-hidden":"true"},w={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},H={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"5.518ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 2439 683","aria-hidden":"true"};function F(v,s,L,C,D,M){return n(),t("div",null,[s[35]||(s[35]=e(`

    Extracting harmonic equations ​

    Harmonic Balance method ​

    Once a DifferentialEquation is defined and its harmonics specified, one can extract the harmonic equations using get_harmonic_equations, which itself is composed of the subroutines harmonic_ansatz, slow_flow, fourier_transform! and drop_powers.

    The harmonic equations use an additional time variable specified as slow_time in get_harmonic_equations. This is essentially a label distinguishing the time dependence of the harmonic variables (expected to be slow) from that of the oscillating terms (expeted to be fast). When the equations are Fourier-transformed to remove oscillating terms, slow_time is treated as a constant. Such an approach is exact when looking for steady states.

    # HarmonicBalance.get_harmonic_equations — Function.
    julia
    get_harmonic_equations(diff_eom::DifferentialEquation; fast_time=nothing, slow_time=nothing)

    Apply the harmonic ansatz, followed by the slow-flow, Fourier transform and dropping higher-order derivatives to obtain a set of ODEs (the harmonic equations) governing the harmonics of diff_eom.

    The harmonics evolve in slow_time, the oscillating terms themselves in fast_time. If no input is used, a variable T is defined for slow_time and fast_time is taken as the independent variable of diff_eom.

    By default, all products of order > 1 of slow_time-derivatives are dropped, which means the equations are linear in the time-derivatives.

    Example

    julia
    julia> @variables t, x(t), ω0, ω, F;
+
+# enter the simple harmonic oscillator
+julia> diff_eom = DifferentialEquation( d(x,t,2) + ω0^2 * x ~ F *cos(ω*t), x);
+
+# expand x in the harmonic ω
+julia> add_harmonic!(diff_eom, x, ω);
+
+# get equations for the harmonics evolving in the slow time T
+julia> harmonic_eom = get_harmonic_equations(diff_eom)
+
+A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω0, ω, F
+
+Harmonic ansatz:
+x(t) = u1*cos(ωt) + v1*sin(ωt)
+
+Harmonic equations:
+
+(ω0^2)*u1(T) + (2//1)*ω*Differential(T)(v1(T)) - (ω^2)*u1(T) ~ F
+
+(ω0^2)*v1(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) ~ 0

    source


    # HarmonicBalance.harmonic_ansatz — Function.
    julia
    harmonic_ansatz(eom::DifferentialEquation, time::Num; coordinates="Cartesian")

    Expand each variable of diff_eom using the harmonics assigned to it with time as the time variable. For each harmonic of each variable, instance(s) of HarmonicVariable are automatically created and named.

    source


    # HarmonicBalance.slow_flow — Function.
    julia
    slow_flow(eom::HarmonicEquation; fast_time::Num, slow_time::Num, degree=2)

    Removes all derivatives w.r.t fast_time (and their products) in eom of power degree. In the remaining derivatives, fast_time is replaced by slow_time.

    source


    # HarmonicBalance.fourier_transform — Function.
    julia
    fourier_transform(
+    eom::HarmonicEquation,
+    time::Num
+) -> HarmonicEquation

    Extract the Fourier components of eom corresponding to the harmonics specified in eom.variables. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated time does not appear in the resulting equations anymore.

    Underlying assumption: all time-dependences are harmonic.

    source


    # HarmonicBalance.drop_powers — Function.
    julia
    drop_powers(expr, vars, deg)

    Remove parts of expr where the combined power of vars is => deg.

    Example

    julia
    julia> @variables x,y;
+julia>drop_powers((x+y)^2, x, 2)
+y^2 + 2*x*y
+julia>drop_powers((x+y)^2, [x,y], 2)
+0
+julia>drop_powers((x+y)^2 + (x+y)^3, [x,y], 3)
+x^2 + y^2 + 2*x*y

    source


    HarmonicVariable and HarmonicEquation types ​

    `,15)),a("p",null,[s[10]||(s[10]=i("The equations governing the harmonics are stored using the two following structs. When going from the original to the harmonic equations, the harmonic ansatz ")),a("mjx-container",o,[(n(),t("svg",Q,s[0]||(s[0]=[e('',1)]))),s[1]||(s[1]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("msub",null,[a("mi",null,"x"),a("mi",null,"i")]),a("mo",{stretchy:"false"},"("),a("mi",null,"t"),a("mo",{stretchy:"false"},")"),a("mo",null,"="),a("munderover",null,[a("mo",{"data-mjx-texclass":"OP"},"∑"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"j"),a("mo",null,"="),a("mn",null,"1")]),a("mi",null,"M")]),a("msub",null,[a("mi",null,"u"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mo",{stretchy:"false"},"("),a("mi",null,"T"),a("mo",{stretchy:"false"},")"),a("mi",null,"cos"),a("mo",{"data-mjx-texclass":"NONE"},"⁡"),a("mo",{stretchy:"false"},"("),a("msub",null,[a("mi",null,"ω"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mi",null,"t"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("msub",null,[a("mi",null,"v"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mo",{stretchy:"false"},"("),a("mi",null,"T"),a("mo",{stretchy:"false"},")"),a("mi",null,"sin"),a("mo",{"data-mjx-texclass":"NONE"},"⁡"),a("mo",{stretchy:"false"},"("),a("msub",null,[a("mi",null,"ω"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mi",null,"t"),a("mo",{stretchy:"false"},")")])],-1))]),s[11]||(s[11]=i(" is used. Internally, each pair ")),a("mjx-container",T,[(n(),t("svg",h,s[2]||(s[2]=[e('',1)]))),s[3]||(s[3]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mo",{stretchy:"false"},"("),a("msub",null,[a("mi",null,"u"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mo",null,","),a("msub",null,[a("mi",null,"v"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mo",{stretchy:"false"},")")])],-1))]),s[12]||(s[12]=i(" is stored as a ")),s[13]||(s[13]=a("code",null,"HarmonicVariable",-1)),s[14]||(s[14]=i(". 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    # HarmonicBalance.HarmonicVariable — Type.
    julia
    mutable struct HarmonicVariable

    Holds a variable stored under symbol describing the harmonic ω of natural_variable.

    Fields

    • symbol::Num: Symbol of the variable in the HarmonicBalance namespace.

    • name::String: Human-readable labels of the variable, used for plotting.

    • type::String: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)

    • ω::Num: The harmonic being described.

    • natural_variable::Num: The natural variable whose harmonic is being described.

    source


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    A HarmonicEquation can be either parsed into a steady-state Problem or solved using a dynamical ODE solver.

    # HarmonicBalance.HarmonicEquation — Type.
    julia
    mutable struct HarmonicEquation

    Holds a set of algebraic equations governing the harmonics of a DifferentialEquation.

    Fields

    • equations::Vector{Equation}: A set of equations governing the harmonics.

    • variables::Vector{HarmonicVariable}: A set of variables describing the harmonics.

    • parameters::Vector{Num}: The parameters of the equation set.

    • natural_equation::DifferentialEquation: The natural equation (before the harmonic ansatz was used).

    source


    ',3))])}const A=l(r,[["render",F]]);export{Z as __pageData,A as default}; diff --git a/v0.10.7/assets/manual_extracting_harmonics.md.euttv_oy.lean.js b/v0.10.7/assets/manual_extracting_harmonics.md.euttv_oy.lean.js new file mode 100644 index 00000000..38f19ce5 --- /dev/null +++ b/v0.10.7/assets/manual_extracting_harmonics.md.euttv_oy.lean.js @@ -0,0 +1,32 @@ +import{_ as l,c as t,a4 as e,j as a,a as i,o as n}from"./chunks/framework.D3a-l5sU.js";const Z=JSON.parse('{"title":"Extracting harmonic equations","description":"","frontmatter":{},"headers":[],"relativePath":"manual/extracting_harmonics.md","filePath":"manual/extracting_harmonics.md"}'),r={name:"manual/extracting_harmonics.md"},o={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},Q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.972ex"},xmlns:"http://www.w3.org/2000/svg",width:"47.051ex",height:"3.144ex",role:"img",focusable:"false",viewBox:"0 -960 20796.4 1389.6","aria-hidden":"true"},T={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},h={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.666ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.845ex",height:"2.363ex",role:"img",focusable:"false",viewBox:"0 -750 3909.4 1044.2","aria-hidden":"true"},d={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.666ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.251ex",height:"1.668ex",role:"img",focusable:"false",viewBox:"0 -443 1436.9 737.2","aria-hidden":"true"},k={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.611ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2038 1000","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.611ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2038 1000","aria-hidden":"true"},u={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.378ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 1051 683","aria-hidden":"true"},E={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},x={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.009ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 888 683","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},f={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"5.518ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 2439 683","aria-hidden":"true"},w={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},H={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"5.518ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 2439 683","aria-hidden":"true"};function F(v,s,L,C,D,M){return n(),t("div",null,[s[35]||(s[35]=e(`

    Extracting harmonic equations ​

    Harmonic Balance method ​

    Once a DifferentialEquation is defined and its harmonics specified, one can extract the harmonic equations using get_harmonic_equations, which itself is composed of the subroutines harmonic_ansatz, slow_flow, fourier_transform! and drop_powers.

    The harmonic equations use an additional time variable specified as slow_time in get_harmonic_equations. This is essentially a label distinguishing the time dependence of the harmonic variables (expected to be slow) from that of the oscillating terms (expeted to be fast). When the equations are Fourier-transformed to remove oscillating terms, slow_time is treated as a constant. Such an approach is exact when looking for steady states.

    # HarmonicBalance.get_harmonic_equations — Function.
    julia
    get_harmonic_equations(diff_eom::DifferentialEquation; fast_time=nothing, slow_time=nothing)

    Apply the harmonic ansatz, followed by the slow-flow, Fourier transform and dropping higher-order derivatives to obtain a set of ODEs (the harmonic equations) governing the harmonics of diff_eom.

    The harmonics evolve in slow_time, the oscillating terms themselves in fast_time. If no input is used, a variable T is defined for slow_time and fast_time is taken as the independent variable of diff_eom.

    By default, all products of order > 1 of slow_time-derivatives are dropped, which means the equations are linear in the time-derivatives.

    Example

    julia
    julia> @variables t, x(t), ω0, ω, F;
+
+# enter the simple harmonic oscillator
+julia> diff_eom = DifferentialEquation( d(x,t,2) + ω0^2 * x ~ F *cos(ω*t), x);
+
+# expand x in the harmonic ω
+julia> add_harmonic!(diff_eom, x, ω);
+
+# get equations for the harmonics evolving in the slow time T
+julia> harmonic_eom = get_harmonic_equations(diff_eom)
+
+A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω0, ω, F
+
+Harmonic ansatz:
+x(t) = u1*cos(ωt) + v1*sin(ωt)
+
+Harmonic equations:
+
+(ω0^2)*u1(T) + (2//1)*ω*Differential(T)(v1(T)) - (ω^2)*u1(T) ~ F
+
+(ω0^2)*v1(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) ~ 0

    source


    # HarmonicBalance.harmonic_ansatz — Function.
    julia
    harmonic_ansatz(eom::DifferentialEquation, time::Num; coordinates="Cartesian")

    Expand each variable of diff_eom using the harmonics assigned to it with time as the time variable. For each harmonic of each variable, instance(s) of HarmonicVariable are automatically created and named.

    source


    # HarmonicBalance.slow_flow — Function.
    julia
    slow_flow(eom::HarmonicEquation; fast_time::Num, slow_time::Num, degree=2)

    Removes all derivatives w.r.t fast_time (and their products) in eom of power degree. In the remaining derivatives, fast_time is replaced by slow_time.

    source


    # HarmonicBalance.fourier_transform — Function.
    julia
    fourier_transform(
+    eom::HarmonicEquation,
+    time::Num
+) -> HarmonicEquation

    Extract the Fourier components of eom corresponding to the harmonics specified in eom.variables. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated time does not appear in the resulting equations anymore.

    Underlying assumption: all time-dependences are harmonic.

    source


    # HarmonicBalance.drop_powers — Function.
    julia
    drop_powers(expr, vars, deg)

    Remove parts of expr where the combined power of vars is => deg.

    Example

    julia
    julia> @variables x,y;
+julia>drop_powers((x+y)^2, x, 2)
+y^2 + 2*x*y
+julia>drop_powers((x+y)^2, [x,y], 2)
+0
+julia>drop_powers((x+y)^2 + (x+y)^3, [x,y], 3)
+x^2 + y^2 + 2*x*y

    source


    HarmonicVariable and HarmonicEquation types ​

    `,15)),a("p",null,[s[10]||(s[10]=i("The equations governing the harmonics are stored using the two following structs. When going from the original to the harmonic equations, the harmonic ansatz ")),a("mjx-container",o,[(n(),t("svg",Q,s[0]||(s[0]=[e('',1)]))),s[1]||(s[1]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("msub",null,[a("mi",null,"x"),a("mi",null,"i")]),a("mo",{stretchy:"false"},"("),a("mi",null,"t"),a("mo",{stretchy:"false"},")"),a("mo",null,"="),a("munderover",null,[a("mo",{"data-mjx-texclass":"OP"},"∑"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"j"),a("mo",null,"="),a("mn",null,"1")]),a("mi",null,"M")]),a("msub",null,[a("mi",null,"u"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mo",{stretchy:"false"},"("),a("mi",null,"T"),a("mo",{stretchy:"false"},")"),a("mi",null,"cos"),a("mo",{"data-mjx-texclass":"NONE"},"⁡"),a("mo",{stretchy:"false"},"("),a("msub",null,[a("mi",null,"ω"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mi",null,"t"),a("mo",{stretchy:"false"},")"),a("mo",null,"+"),a("msub",null,[a("mi",null,"v"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mo",{stretchy:"false"},"("),a("mi",null,"T"),a("mo",{stretchy:"false"},")"),a("mi",null,"sin"),a("mo",{"data-mjx-texclass":"NONE"},"⁡"),a("mo",{stretchy:"false"},"("),a("msub",null,[a("mi",null,"ω"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mi",null,"t"),a("mo",{stretchy:"false"},")")])],-1))]),s[11]||(s[11]=i(" is used. Internally, each pair ")),a("mjx-container",T,[(n(),t("svg",h,s[2]||(s[2]=[e('',1)]))),s[3]||(s[3]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mo",{stretchy:"false"},"("),a("msub",null,[a("mi",null,"u"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mo",null,","),a("msub",null,[a("mi",null,"v"),a("mrow",{"data-mjx-texclass":"ORD"},[a("mi",null,"i"),a("mo",null,","),a("mi",null,"j")])]),a("mo",{stretchy:"false"},")")])],-1))]),s[12]||(s[12]=i(" is stored as a ")),s[13]||(s[13]=a("code",null,"HarmonicVariable",-1)),s[14]||(s[14]=i(". 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    # HarmonicBalance.HarmonicVariable — Type.
    julia
    mutable struct HarmonicVariable

    Holds a variable stored under symbol describing the harmonic ω of natural_variable.

    Fields

    • symbol::Num: Symbol of the variable in the HarmonicBalance namespace.

    • name::String: Human-readable labels of the variable, used for plotting.

    • type::String: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)

    • ω::Num: The harmonic being described.

    • natural_variable::Num: The natural variable whose harmonic is being described.

    source


    ',2)),a("p",null,[s[26]||(s[26]=i("When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a ")),s[27]||(s[27]=a("code",null,"HarmonicEquation",-1)),s[28]||(s[28]=i(". For an initial equation of motion consisting of ")),a("mjx-container",u,[(n(),t("svg",y,s[18]||(s[18]=[a("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[a("g",{"data-mml-node":"math"},[a("g",{"data-mml-node":"mi"},[a("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),s[19]||(s[19]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mi",null,"M")])],-1))]),s[29]||(s[29]=i(" variables, each expanded in ")),a("mjx-container",E,[(n(),t("svg",x,s[20]||(s[20]=[a("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[a("g",{"data-mml-node":"math"},[a("g",{"data-mml-node":"mi"},[a("path",{"data-c":"1D441",d:"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[21]||(s[21]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mi",null,"N")])],-1))]),s[30]||(s[30]=i(" harmonics, the resulting ")),s[31]||(s[31]=a("code",null,"HarmonicEquation",-1)),s[32]||(s[32]=i(" holds ")),a("mjx-container",b,[(n(),t("svg",f,s[22]||(s[22]=[e('',1)]))),s[23]||(s[23]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mn",null,"2"),a("mi",null,"N"),a("mi",null,"M")])],-1))]),s[33]||(s[33]=i(" equations of ")),a("mjx-container",w,[(n(),t("svg",H,s[24]||(s[24]=[e('',1)]))),s[25]||(s[25]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mn",null,"2"),a("mi",null,"N"),a("mi",null,"M")])],-1))]),s[34]||(s[34]=i(" variables. Each symbol not corresponding to a variable is identified as a parameter."))]),s[37]||(s[37]=e('

    A HarmonicEquation can be either parsed into a steady-state Problem or solved using a dynamical ODE solver.

    # HarmonicBalance.HarmonicEquation — Type.
    julia
    mutable struct HarmonicEquation

    Holds a set of algebraic equations governing the harmonics of a DifferentialEquation.

    Fields

    • equations::Vector{Equation}: A set of equations governing the harmonics.

    • variables::Vector{HarmonicVariable}: A set of variables describing the harmonics.

    • parameters::Vector{Num}: The parameters of the equation set.

    • natural_equation::DifferentialEquation: The natural equation (before the harmonic ansatz was used).

    source


    ',3))])}const A=l(r,[["render",F]]);export{Z as __pageData,A as default}; diff --git a/v0.10.7/assets/manual_linear_response.md.BVEaw-Id.js b/v0.10.7/assets/manual_linear_response.md.BVEaw-Id.js new file mode 100644 index 00000000..5315a7fa --- /dev/null +++ b/v0.10.7/assets/manual_linear_response.md.BVEaw-Id.js @@ -0,0 +1,5 @@ +import{_ as r,c as s,a4 as i,j as a,a as t,o as n}from"./chunks/framework.D3a-l5sU.js";const x=JSON.parse('{"title":"Linear response (WIP)","description":"","frontmatter":{},"headers":[],"relativePath":"manual/linear_response.md","filePath":"manual/linear_response.md"}'),o={name:"manual/linear_response.md"},l={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"},d={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},h={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.247ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2319 1000","aria-hidden":"true"},c={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.278ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2333 1000","aria-hidden":"true"};function m(g,e,b,u,y,f){return n(),s("div",null,[e[10]||(e[10]=i('

    Linear response (WIP) ​

    This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.

    The methodology used is explained in Jan Kosata phd thesis.

    Stability ​

    The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,

    # HarmonicBalance.LinearResponse.get_Jacobian — Function.
    julia
    get_Jacobian(eom)

    Obtain the symbolic Jacobian matrix of eom (either a HarmonicEquation or a DifferentialEquation). This is the linearised left-hand side of F(u) = du/dT.

    source

    Obtain a Jacobian from a DifferentialEquation by first converting it into a HarmonicEquation.

    source

    Get the Jacobian of a set of equations eqs with respect to the variables vars.

    source


    Linear response ​

    The response to white noise can be shown with plot_linear_response. Depending on the order argument, different methods are used.

    # HarmonicBalance.LinearResponse.plot_linear_response — Function.
    julia
    plot_linear_response(res::Result, nat_var::Num; Ω_range, branch::Int, order=1, logscale=false, show_progress=true, kwargs...)

    Plot the linear response to white noise of the variable nat_var for Result res on branch for input frequencies Ω_range. Slow-time derivatives up to order are kept in the process.

    Any kwargs are fed to Plots' gr().

    Solutions not belonging to the physical class are ignored.

    source


    First order ​

    ',12)),a("p",null,[e[6]||(e[6]=t("The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues ")),a("mjx-container",l,[(n(),s("svg",p,e[0]||(e[0]=[a("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[a("g",{"data-mml-node":"math"},[a("g",{"data-mml-node":"mi"},[a("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),e[1]||(e[1]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mi",null,"λ")])],-1))]),e[7]||(e[7]=t(" describes a Lorentzian peak in the response; ")),a("mjx-container",d,[(n(),s("svg",h,e[2]||(e[2]=[i('',1)]))),e[3]||(e[3]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mtext",null,"Re"),a("mo",{stretchy:"false"},"["),a("mi",null,"λ"),a("mo",{stretchy:"false"},"]")])],-1))]),e[8]||(e[8]=t(" gives its center and ")),a("mjx-container",c,[(n(),s("svg",k,e[4]||(e[4]=[i('',1)]))),e[5]||(e[5]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mtext",null,"Im"),a("mo",{stretchy:"false"},"["),a("mi",null,"λ"),a("mo",{stretchy:"false"},"]")])],-1))]),e[9]||(e[9]=t(" its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment."))]),e[11]||(e[11]=i(`

    The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).

    Behind the scenes, the spectra are stored using the dedicated structs Lorentzian and JacobianSpectrum.

    # HarmonicBalance.LinearResponse.JacobianSpectrum — Type.
    julia
    mutable struct JacobianSpectrum

    Holds a set of Lorentzian objects belonging to a variable.

    Fields

    • peaks::Vector{HarmonicBalance.LinearResponse.Lorentzian}

    Constructor

    julia
    JacobianSpectrum(res::Result; index::Int, branch::Int)

    source


    # HarmonicBalance.LinearResponse.Lorentzian — Type.
    julia
    struct Lorentzian

    Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).

    Fields

    • ω0::Float64

    • Γ::Float64

    • A::Float64

    source


    Higher orders ​

    Setting order > 1 increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.

    # HarmonicBalance.LinearResponse.ResponseMatrix — Type.
    julia
    struct ResponseMatrix

    Holds the compiled response matrix of a system.

    Fields

    • matrix::Matrix{Function}: The response matrix (compiled).

    • symbols::Vector{Num}: Any symbolic variables in matrix to be substituted at evaluation.

    • variables::Vector{HarmonicVariable}: The frequencies of the harmonic variables underlying matrix. These are needed to transform the harmonic variables to the non-rotating frame.

    source


    # HarmonicBalance.LinearResponse.get_response — Function.
    julia
    get_response(
+    rmat::HarmonicBalance.LinearResponse.ResponseMatrix,
+    s::OrderedCollections.OrderedDict{Num, ComplexF64},
+    Ω
+) -> Any

    For rmat and a solution dictionary s, calculate the total response to a perturbative force at frequency Ω.

    source


    # HarmonicBalance.LinearResponse.get_response_matrix — Function.
    julia
    get_response_matrix(diff_eq::DifferentialEquation, freq::Num; order=2)

    Obtain the symbolic linear response matrix of a diff_eq corresponding to a perturbation frequency freq. This routine cannot accept a HarmonicEquation since there, some time-derivatives are already dropped. order denotes the highest differential order to be considered.

    source


    `,14))])}const Q=r(o,[["render",m]]);export{x as __pageData,Q as default}; diff --git a/v0.10.7/assets/manual_linear_response.md.BVEaw-Id.lean.js b/v0.10.7/assets/manual_linear_response.md.BVEaw-Id.lean.js new file mode 100644 index 00000000..5315a7fa --- /dev/null +++ b/v0.10.7/assets/manual_linear_response.md.BVEaw-Id.lean.js @@ -0,0 +1,5 @@ +import{_ as r,c as s,a4 as i,j as a,a as t,o as n}from"./chunks/framework.D3a-l5sU.js";const x=JSON.parse('{"title":"Linear response (WIP)","description":"","frontmatter":{},"headers":[],"relativePath":"manual/linear_response.md","filePath":"manual/linear_response.md"}'),o={name:"manual/linear_response.md"},l={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"},d={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},h={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.247ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2319 1000","aria-hidden":"true"},c={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.278ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2333 1000","aria-hidden":"true"};function m(g,e,b,u,y,f){return n(),s("div",null,[e[10]||(e[10]=i('

    Linear response (WIP) ​

    This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.

    The methodology used is explained in Jan Kosata phd thesis.

    Stability ​

    The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,

    # HarmonicBalance.LinearResponse.get_Jacobian — Function.
    julia
    get_Jacobian(eom)

    Obtain the symbolic Jacobian matrix of eom (either a HarmonicEquation or a DifferentialEquation). This is the linearised left-hand side of F(u) = du/dT.

    source

    Obtain a Jacobian from a DifferentialEquation by first converting it into a HarmonicEquation.

    source

    Get the Jacobian of a set of equations eqs with respect to the variables vars.

    source


    Linear response ​

    The response to white noise can be shown with plot_linear_response. Depending on the order argument, different methods are used.

    # HarmonicBalance.LinearResponse.plot_linear_response — Function.
    julia
    plot_linear_response(res::Result, nat_var::Num; Ω_range, branch::Int, order=1, logscale=false, show_progress=true, kwargs...)

    Plot the linear response to white noise of the variable nat_var for Result res on branch for input frequencies Ω_range. Slow-time derivatives up to order are kept in the process.

    Any kwargs are fed to Plots' gr().

    Solutions not belonging to the physical class are ignored.

    source


    First order ​

    ',12)),a("p",null,[e[6]||(e[6]=t("The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues ")),a("mjx-container",l,[(n(),s("svg",p,e[0]||(e[0]=[a("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[a("g",{"data-mml-node":"math"},[a("g",{"data-mml-node":"mi"},[a("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),e[1]||(e[1]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mi",null,"λ")])],-1))]),e[7]||(e[7]=t(" describes a Lorentzian peak in the response; ")),a("mjx-container",d,[(n(),s("svg",h,e[2]||(e[2]=[i('',1)]))),e[3]||(e[3]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mtext",null,"Re"),a("mo",{stretchy:"false"},"["),a("mi",null,"λ"),a("mo",{stretchy:"false"},"]")])],-1))]),e[8]||(e[8]=t(" gives its center and ")),a("mjx-container",c,[(n(),s("svg",k,e[4]||(e[4]=[i('',1)]))),e[5]||(e[5]=a("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[a("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[a("mtext",null,"Im"),a("mo",{stretchy:"false"},"["),a("mi",null,"λ"),a("mo",{stretchy:"false"},"]")])],-1))]),e[9]||(e[9]=t(" its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment."))]),e[11]||(e[11]=i(`

    The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).

    Behind the scenes, the spectra are stored using the dedicated structs Lorentzian and JacobianSpectrum.

    # HarmonicBalance.LinearResponse.JacobianSpectrum — Type.
    julia
    mutable struct JacobianSpectrum

    Holds a set of Lorentzian objects belonging to a variable.

    Fields

    • peaks::Vector{HarmonicBalance.LinearResponse.Lorentzian}

    Constructor

    julia
    JacobianSpectrum(res::Result; index::Int, branch::Int)

    source


    # HarmonicBalance.LinearResponse.Lorentzian — Type.
    julia
    struct Lorentzian

    Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).

    Fields

    • ω0::Float64

    • Γ::Float64

    • A::Float64

    source


    Higher orders ​

    Setting order > 1 increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.

    # HarmonicBalance.LinearResponse.ResponseMatrix — Type.
    julia
    struct ResponseMatrix

    Holds the compiled response matrix of a system.

    Fields

    • matrix::Matrix{Function}: The response matrix (compiled).

    • symbols::Vector{Num}: Any symbolic variables in matrix to be substituted at evaluation.

    • variables::Vector{HarmonicVariable}: The frequencies of the harmonic variables underlying matrix. These are needed to transform the harmonic variables to the non-rotating frame.

    source


    # HarmonicBalance.LinearResponse.get_response — Function.
    julia
    get_response(
+    rmat::HarmonicBalance.LinearResponse.ResponseMatrix,
+    s::OrderedCollections.OrderedDict{Num, ComplexF64},
+    Ω
+) -> Any

    For rmat and a solution dictionary s, calculate the total response to a perturbative force at frequency Ω.

    source


    # HarmonicBalance.LinearResponse.get_response_matrix — Function.
    julia
    get_response_matrix(diff_eq::DifferentialEquation, freq::Num; order=2)

    Obtain the symbolic linear response matrix of a diff_eq corresponding to a perturbation frequency freq. This routine cannot accept a HarmonicEquation since there, some time-derivatives are already dropped. order denotes the highest differential order to be considered.

    source


    `,14))])}const Q=r(o,[["render",m]]);export{x as __pageData,Q as default}; diff --git a/v0.10.7/assets/manual_plotting.md.CxPD2Nsz.js b/v0.10.7/assets/manual_plotting.md.CxPD2Nsz.js new file mode 100644 index 00000000..f9a9a818 --- /dev/null +++ b/v0.10.7/assets/manual_plotting.md.CxPD2Nsz.js @@ -0,0 +1,10 @@ +import{_ as a,c as t,a4 as i,o as e}from"./chunks/framework.D3a-l5sU.js";const g=JSON.parse('{"title":"Analysis and plotting","description":"","frontmatter":{},"headers":[],"relativePath":"manual/plotting.md","filePath":"manual/plotting.md"}'),n={name:"manual/plotting.md"};function l(o,s,p,r,d,h){return e(),t("div",null,s[0]||(s[0]=[i(`

    Analysis and plotting ​

    The key method for visualization is transform_solutions, which parses a string into a symbolic expression and evaluates it for every steady state solution.

    # HarmonicBalance.transform_solutions — Function.
    julia
    transform_solutions(
+    res::Result,
+    func;
+    branches,
+    realify
+) -> Vector

    Takes a Result object and a string f representing a Symbolics.jl expression. Returns an array with the values of f evaluated for the respective solutions. Additional substitution rules can be specified in rules in the format ("a" => val) or (a => val)

    source


    Plotting solutions ​

    The function plot is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by sort_solutions.

    # RecipesBase.plot — Method.
    julia
    plot(res::Result, varargs...; cut, kwargs...) -> Plots.Plot

    Plot a Result object.

    Class selection done by passing String or Vector{String} as kwarg:

    class       :   only plot solutions in this class(es) ("all" --> plot everything)
+not_class   :   do not plot solutions in this class(es)

    Other kwargs are passed onto Plots.gr().

    See also plot!

    The x,y,z arguments are Strings compatible with Symbolics.jl, e.g., y=2*sqrt(u1^2+v1^2) plots the amplitude of the first quadratures multiplied by 2.

    1D plots

    plot(res::Result; x::String, y::String, class="default", not_class=[], kwargs...)
+plot(res::Result, y::String; kwargs...) # take x automatically from Result

    Default behaviour is to plot stable solutions as full lines, unstable as dashed.

    If a sweep in two parameters were done, i.e., dim(res)==2, a one dimensional cut can be plotted by using the keyword cut were it takes a Pair{Num, Float64} type entry. For example, plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2)) plots a cut at λ = 0.2.

    2D plots

    plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)

    To make the 2d plot less chaotic it is required to specify the specific branch to plot, labeled by a Int64.

    The x and y axes are taken automatically from res

    source


    Plotting phase diagrams ​

    In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. plot_phase_diagram handles this for 1D and 2D datasets.

    # HarmonicBalance.plot_phase_diagram — Function.
    julia
    plot_phase_diagram(res::Result; kwargs...) -> Plots.Plot

    Plot the number of solutions in a Result object as a function of the parameters. Works with 1D and 2D datasets.

    Class selection done by passing String or Vector{String} as kwarg:

    class::String       :   only count solutions in this class ("all" --> plot everything)
+not_class::String   :   do not count solutions in this class

    Other kwargs are passed onto Plots.gr()

    source


    Plot spaghetti plot ​

    Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with plot_spaghetti.

    # HarmonicBalance.plot_spaghetti — Function.
    julia
    plot_spaghetti(res::Result; x, y, z, kwargs...)

    Plot a three dimension line plot of a Result object as a function of the parameters. Works with 1D and 2D datasets.

    Class selection done by passing String or Vector{String} as kwarg:

    class::String       :   only count solutions in this class ("all" --> plot everything)
+not_class::String   :   do not count solutions in this class

    Other kwargs are passed onto Plots.gr()

    source


    `,16)]))}const u=a(n,[["render",l]]);export{g as __pageData,u as default}; diff --git a/v0.10.7/assets/manual_plotting.md.CxPD2Nsz.lean.js b/v0.10.7/assets/manual_plotting.md.CxPD2Nsz.lean.js new file mode 100644 index 00000000..f9a9a818 --- /dev/null +++ b/v0.10.7/assets/manual_plotting.md.CxPD2Nsz.lean.js @@ -0,0 +1,10 @@ +import{_ as a,c as t,a4 as i,o as e}from"./chunks/framework.D3a-l5sU.js";const g=JSON.parse('{"title":"Analysis and plotting","description":"","frontmatter":{},"headers":[],"relativePath":"manual/plotting.md","filePath":"manual/plotting.md"}'),n={name:"manual/plotting.md"};function l(o,s,p,r,d,h){return e(),t("div",null,s[0]||(s[0]=[i(`

    Analysis and plotting ​

    The key method for visualization is transform_solutions, which parses a string into a symbolic expression and evaluates it for every steady state solution.

    # HarmonicBalance.transform_solutions — Function.
    julia
    transform_solutions(
+    res::Result,
+    func;
+    branches,
+    realify
+) -> Vector

    Takes a Result object and a string f representing a Symbolics.jl expression. Returns an array with the values of f evaluated for the respective solutions. Additional substitution rules can be specified in rules in the format ("a" => val) or (a => val)

    source


    Plotting solutions ​

    The function plot is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by sort_solutions.

    # RecipesBase.plot — Method.
    julia
    plot(res::Result, varargs...; cut, kwargs...) -> Plots.Plot

    Plot a Result object.

    Class selection done by passing String or Vector{String} as kwarg:

    class       :   only plot solutions in this class(es) ("all" --> plot everything)
+not_class   :   do not plot solutions in this class(es)

    Other kwargs are passed onto Plots.gr().

    See also plot!

    The x,y,z arguments are Strings compatible with Symbolics.jl, e.g., y=2*sqrt(u1^2+v1^2) plots the amplitude of the first quadratures multiplied by 2.

    1D plots

    plot(res::Result; x::String, y::String, class="default", not_class=[], kwargs...)
+plot(res::Result, y::String; kwargs...) # take x automatically from Result

    Default behaviour is to plot stable solutions as full lines, unstable as dashed.

    If a sweep in two parameters were done, i.e., dim(res)==2, a one dimensional cut can be plotted by using the keyword cut were it takes a Pair{Num, Float64} type entry. For example, plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2)) plots a cut at λ = 0.2.

    2D plots

    plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)

    To make the 2d plot less chaotic it is required to specify the specific branch to plot, labeled by a Int64.

    The x and y axes are taken automatically from res

    source


    Plotting phase diagrams ​

    In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. plot_phase_diagram handles this for 1D and 2D datasets.

    # HarmonicBalance.plot_phase_diagram — Function.
    julia
    plot_phase_diagram(res::Result; kwargs...) -> Plots.Plot

    Plot the number of solutions in a Result object as a function of the parameters. Works with 1D and 2D datasets.

    Class selection done by passing String or Vector{String} as kwarg:

    class::String       :   only count solutions in this class ("all" --> plot everything)
+not_class::String   :   do not count solutions in this class

    Other kwargs are passed onto Plots.gr()

    source


    Plot spaghetti plot ​

    Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with plot_spaghetti.

    # HarmonicBalance.plot_spaghetti — Function.
    julia
    plot_spaghetti(res::Result; x, y, z, kwargs...)

    Plot a three dimension line plot of a Result object as a function of the parameters. Works with 1D and 2D datasets.

    Class selection done by passing String or Vector{String} as kwarg:

    class::String       :   only count solutions in this class ("all" --> plot everything)
+not_class::String   :   do not count solutions in this class

    Other kwargs are passed onto Plots.gr()

    source


    `,16)]))}const u=a(n,[["render",l]]);export{g as __pageData,u as default}; diff --git a/v0.10.7/assets/manual_saving.md.Bisr1Re6.js b/v0.10.7/assets/manual_saving.md.Bisr1Re6.js new file mode 100644 index 00000000..bcaadb56 --- /dev/null +++ b/v0.10.7/assets/manual_saving.md.Bisr1Re6.js @@ -0,0 +1 @@ +import{_ as a,c as i,a4 as s,o as n}from"./chunks/framework.D3a-l5sU.js";const b=JSON.parse('{"title":"Saving and loading","description":"","frontmatter":{},"headers":[],"relativePath":"manual/saving.md","filePath":"manual/saving.md"}'),o={name:"manual/saving.md"};function t(c,e,d,l,r,p){return n(),i("div",null,e[0]||(e[0]=[s('

    Saving and loading ​

    All of the types native to HarmonicBalance.jl can be saved into a .jld2 file using save and loaded using load. Most of the saving/loading is performed using the package JLD2.jl, with the addition of reinstating the symbolic variables in the HarmonicBalance namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as Result can be saved and loaded with no loss of information.

    The function export_csv saves a .csv file which can be plot elsewhere.

    # HarmonicBalance.save — Function.
    julia
    save(filename, object)

    Saves object into .jld2 file filename (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.

    source


    # HarmonicBalance.load — Function.
    julia
    load(filename)

    Loads an object from filename. For objects containing symbolic expressions such as HarmonicEquation, the symbolic variables are reinstated in the HarmonicBalance namespace.

    source


    # HarmonicBalance.export_csv — Function.
    julia
    export_csv(filename, res, branch)

    Saves into filename a specified solution branch of the Result res.

    source


    ',10)]))}const u=a(o,[["render",t]]);export{b as __pageData,u as default}; diff --git a/v0.10.7/assets/manual_saving.md.Bisr1Re6.lean.js b/v0.10.7/assets/manual_saving.md.Bisr1Re6.lean.js new file mode 100644 index 00000000..bcaadb56 --- /dev/null +++ b/v0.10.7/assets/manual_saving.md.Bisr1Re6.lean.js @@ -0,0 +1 @@ +import{_ as a,c as i,a4 as s,o as n}from"./chunks/framework.D3a-l5sU.js";const b=JSON.parse('{"title":"Saving and loading","description":"","frontmatter":{},"headers":[],"relativePath":"manual/saving.md","filePath":"manual/saving.md"}'),o={name:"manual/saving.md"};function t(c,e,d,l,r,p){return n(),i("div",null,e[0]||(e[0]=[s('

    Saving and loading ​

    All of the types native to HarmonicBalance.jl can be saved into a .jld2 file using save and loaded using load. Most of the saving/loading is performed using the package JLD2.jl, with the addition of reinstating the symbolic variables in the HarmonicBalance namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as Result can be saved and loaded with no loss of information.

    The function export_csv saves a .csv file which can be plot elsewhere.

    # HarmonicBalance.save — Function.
    julia
    save(filename, object)

    Saves object into .jld2 file filename (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.

    source


    # HarmonicBalance.load — Function.
    julia
    load(filename)

    Loads an object from filename. For objects containing symbolic expressions such as HarmonicEquation, the symbolic variables are reinstated in the HarmonicBalance namespace.

    source


    # HarmonicBalance.export_csv — Function.
    julia
    export_csv(filename, res, branch)

    Saves into filename a specified solution branch of the Result res.

    source


    ',10)]))}const u=a(o,[["render",t]]);export{b as __pageData,u as default}; diff --git a/v0.10.7/assets/manual_solving_harmonics.md.BFw1mOb-.js b/v0.10.7/assets/manual_solving_harmonics.md.BFw1mOb-.js new file mode 100644 index 00000000..b010ed1e --- /dev/null +++ b/v0.10.7/assets/manual_solving_harmonics.md.BFw1mOb-.js @@ -0,0 +1,46 @@ +import{_ as l,c as e,a4 as a,j as s,a as n,o as t}from"./chunks/framework.D3a-l5sU.js";const T=JSON.parse('{"title":"Solving harmonic equations","description":"","frontmatter":{},"headers":[],"relativePath":"manual/solving_harmonics.md","filePath":"manual/solving_harmonics.md"}'),o={name:"manual/solving_harmonics.md"},h={style:{"border-width":"1px","border-style":"solid","border-color":"black",padding:"1em","border-radius":"25px"}},r={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.687ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.124ex",height:"2.573ex",role:"img",focusable:"false",viewBox:"0 -833.9 11988.7 1137.4","aria-hidden":"true"},d={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"};function c(g,i,m,u,E,y){return t(),e("div",null,[i[8]||(i[8]=a(`

    Solving harmonic equations ​

    Once a differential equation of motion has been defined in DifferentialEquation and converted to a HarmonicEquation, we may use the homotopy continuation method (as implemented in HomotopyContinuation.jl) to find steady states. This means that, having called get_harmonic_equations, we need to set all time-derivatives to zero and parse the resulting algebraic equations into a Problem.

    Problem holds the steady-state equations, and (optionally) the symbolic Jacobian which is needed for stability / linear response calculations.

    Once defined, a Problem can be solved for a set of input parameters using get_steady_states to obtain Result.

    # HarmonicBalance.Problem — Type.
    julia
    mutable struct Problem

    Holds a set of algebraic equations describing the steady state of a system.

    Fields

    • variables::Vector{Num}: The harmonic variables to be solved for.

    • parameters::Vector{Num}: All symbols which are not the harmonic variables.

    • system::HomotopyContinuation.ModelKit.System: The input object for HomotopyContinuation.jl solver methods.

    • jacobian::Any: The Jacobian matrix (possibly symbolic). If false, the Jacobian is ignored (may be calculated implicitly after solving).

    • eom::HarmonicEquation: The HarmonicEquation object used to generate this Problem.

    Constructors

    julia
    Problem(eom::HarmonicEquation; Jacobian=true) # find and store the symbolic Jacobian
+Problem(eom::HarmonicEquation; Jacobian="implicit") # ignore the Jacobian for now, compute implicitly later
+Problem(eom::HarmonicEquation; Jacobian=J) # use J as the Jacobian (a function that takes a Dict)
+Problem(eom::HarmonicEquation; Jacobian=false) # ignore the Jacobian

    source


    `,6)),s("div",h,[i[6]||(i[6]=a(`# HarmonicBalance.get_steady_states — Function.
    julia
    get_steady_states(prob::Problem,
+                    swept_parameters::ParameterRange,
+                    fixed_parameters::ParameterList;
+                    method=:warmup,
+                    threading = Threads.nthreads() > 1,
+                    show_progress=true,
+                    sorting="nearest")

    Solves prob over the ranges specified by swept_parameters, keeping fixed_parameters constant. swept_parameters accepts pairs mapping symbolic variables to arrays or LinRange. fixed_parameters accepts pairs mapping symbolic variables to numbers.

    Keyword arguments

    • method: If :warmup (default), a problem similar to prob but with random complex parameters is first solved to find all non-singular paths. The subsequent tracking to find results for all swept_parameters is then much faster than the initial solving. If method=:total_degree, each parameter point is solved separately by tracking the maximum number of paths (employs a total degree homotopy).

    This takes far longer but can be more reliable.

    • threading: If true, multithreaded support is activated. The number of available threads is set by the environment variable JULIA_NUM_THREADS.

    • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".

    • show_progress: Indicate whether a progress bar should be displayed.

    `,12)),s("p",null,[i[4]||(i[4]=n("Example: solving a simple harmonic oscillator ")),s("mjx-container",r,[(t(),e("svg",p,i[0]||(i[0]=[a('',1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"m"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"x"),s("mo",null,"¨")])]),s("mo",null,"+"),s("mi",null,"γ"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"x"),s("mo",null,"˙")])]),s("mo",null,"+"),s("msubsup",null,[s("mi",null,"ω"),s("mn",null,"0"),s("mn",null,"2")]),s("mi",null,"x"),s("mo",null,"="),s("mi",null,"F"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"ω"),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),i[5]||(i[5]=n(" to obtain the response as a function of ")),s("mjx-container",d,[(t(),e("svg",k,i[2]||(i[2]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),i[3]||(i[3]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω")])],-1))])]),i[7]||(i[7]=a(`
    julia
    # having obtained a Problem object, let's find steady states
+julia> range = ParameterRange(ω => LinRange(0.8,1.2,100) ) # 100 parameter sets to solve
+julia> fixed = ParameterList(m => 1, γ => 0.01, F => 0.5, ω_0 => 1)
+julia> get_steady_states(problem, range, fixed)
+
+A steady state result for 100 parameter points
+
+    Solution branches:   1
+       of which real:    1
+       of which stable:  1
+
+    Classes: stable, physical, Hopf, binary_labels

    It is also possible to create multi-dimensional solutions plots.

    julia
    # The swept parameters take precedence over fixed -> use the same fixed
+julia> range = ParameterRange(ω => LinRange(0.8,1.2,100), F => LinRange(0.1,1.0,10) ) # 100x10 parameter sets
+
+# The swept parameters take precedence over fixed -> the F in fixed is now ignored
+julia> get_steady_states(problem, range, fixed)
+
+A steady state result for 1000 parameter points
+
+    Solution branches:   1
+       of which real:    1
+       of which stable:  1
+
+    Classes: stable, physical, Hopf, binary_labels

    source

    `,4))]),i[9]||(i[9]=a(`
    # HarmonicBalance.Result — Type.
    julia
    mutable struct Result

    Stores the steady states of a HarmonicEquation.

    Fields

    • solutions::Array{Vector{Vector{ComplexF64}}}: The variable values of steady-state solutions.

    • swept_parameters::OrderedCollections.OrderedDict{Num, Vector{Union{Float64, ComplexF64}}}: Values of all parameters for all solutions.

    • fixed_parameters::OrderedCollections.OrderedDict{Num, Float64}: The parameters fixed throughout the solutions.

    • problem::Problem: The Problem used to generate this.

    • classes::Dict{String, Array}: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method classify_solutions!).

    • jacobian::Function: The Jacobian with fixed_parameters already substituted. Accepts a dictionary specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function. If problem.jacobian was false, this holds a function that rearranges the equations to find J only after numerical values are inserted (preferable in cases where the symbolic J would be very large).

    • seed::Union{Nothing, UInt32}: Seed used for the solver

    source


    Classifying solutions ​

    The solutions in Result are accompanied by similarly-sized boolean arrays stored in the dictionary Result.classes. The classes can be used by the plotting functions to show/hide/label certain solutions.

    By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with classify_solutions!.

    # HarmonicBalance.classify_solutions! — Function.
    julia
    classify_solutions!(
+    res::Result,
+    func::Union{Function, String},
+    name::String;
+    physical
+) -> Any

    Creates a solution class in res using the inequality condition (parsed into Symbolics.jl input).

    The new class is labelled with name and stored under res.classes[name].

    By default, only physical (=real) solutions are classified, false is returned for the rest.

    Example

    julia
    # solve a previously-defined problem
+res = get_steady_states(problem, swept_parameters, fixed_parameters)
+
+# classify, store in result.classes["large_amplitude"]
+classify_solutions!(res, "sqrt(u1^2 + v1^2) > 1.0" , "large_amplitude")

    source


    Sorting solutions ​

    Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.

    For stable states, the branches describe a system's behaviour under adiabatic parameter changes.

    Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.

    The function sort_solutions goes over the the raw output of get_steady_states and sorts each entry such that neighboring solution sets minimize Euclidean distance.

    Currently, sort_solutions is compatible with 1D and 2D arrays of solution sets.

    # HarmonicBalance.sort_solutions — Function.
    julia
    sort_solutions(
+    solutions::Array;
+    sorting,
+    show_progress
+) -> Any

    Sorts solutions into branches according to the method sorting.

    solutions is an n-dimensional array of Vector{Vector}. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.

    Keyword arguments

    • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".

    • show_progress: Indicate whether a progress bar should be displayed.

    source


    `,16))])}const b=l(o,[["render",c]]);export{T as __pageData,b as default}; diff --git a/v0.10.7/assets/manual_solving_harmonics.md.BFw1mOb-.lean.js b/v0.10.7/assets/manual_solving_harmonics.md.BFw1mOb-.lean.js new file mode 100644 index 00000000..b010ed1e --- /dev/null +++ b/v0.10.7/assets/manual_solving_harmonics.md.BFw1mOb-.lean.js @@ -0,0 +1,46 @@ +import{_ as l,c as e,a4 as a,j as s,a as n,o as t}from"./chunks/framework.D3a-l5sU.js";const T=JSON.parse('{"title":"Solving harmonic equations","description":"","frontmatter":{},"headers":[],"relativePath":"manual/solving_harmonics.md","filePath":"manual/solving_harmonics.md"}'),o={name:"manual/solving_harmonics.md"},h={style:{"border-width":"1px","border-style":"solid","border-color":"black",padding:"1em","border-radius":"25px"}},r={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.687ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.124ex",height:"2.573ex",role:"img",focusable:"false",viewBox:"0 -833.9 11988.7 1137.4","aria-hidden":"true"},d={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"};function c(g,i,m,u,E,y){return t(),e("div",null,[i[8]||(i[8]=a(`

    Solving harmonic equations ​

    Once a differential equation of motion has been defined in DifferentialEquation and converted to a HarmonicEquation, we may use the homotopy continuation method (as implemented in HomotopyContinuation.jl) to find steady states. This means that, having called get_harmonic_equations, we need to set all time-derivatives to zero and parse the resulting algebraic equations into a Problem.

    Problem holds the steady-state equations, and (optionally) the symbolic Jacobian which is needed for stability / linear response calculations.

    Once defined, a Problem can be solved for a set of input parameters using get_steady_states to obtain Result.

    # HarmonicBalance.Problem — Type.
    julia
    mutable struct Problem

    Holds a set of algebraic equations describing the steady state of a system.

    Fields

    • variables::Vector{Num}: The harmonic variables to be solved for.

    • parameters::Vector{Num}: All symbols which are not the harmonic variables.

    • system::HomotopyContinuation.ModelKit.System: The input object for HomotopyContinuation.jl solver methods.

    • jacobian::Any: The Jacobian matrix (possibly symbolic). If false, the Jacobian is ignored (may be calculated implicitly after solving).

    • eom::HarmonicEquation: The HarmonicEquation object used to generate this Problem.

    Constructors

    julia
    Problem(eom::HarmonicEquation; Jacobian=true) # find and store the symbolic Jacobian
+Problem(eom::HarmonicEquation; Jacobian="implicit") # ignore the Jacobian for now, compute implicitly later
+Problem(eom::HarmonicEquation; Jacobian=J) # use J as the Jacobian (a function that takes a Dict)
+Problem(eom::HarmonicEquation; Jacobian=false) # ignore the Jacobian

    source


    `,6)),s("div",h,[i[6]||(i[6]=a(`# HarmonicBalance.get_steady_states — Function.
    julia
    get_steady_states(prob::Problem,
+                    swept_parameters::ParameterRange,
+                    fixed_parameters::ParameterList;
+                    method=:warmup,
+                    threading = Threads.nthreads() > 1,
+                    show_progress=true,
+                    sorting="nearest")

    Solves prob over the ranges specified by swept_parameters, keeping fixed_parameters constant. swept_parameters accepts pairs mapping symbolic variables to arrays or LinRange. fixed_parameters accepts pairs mapping symbolic variables to numbers.

    Keyword arguments

    • method: If :warmup (default), a problem similar to prob but with random complex parameters is first solved to find all non-singular paths. The subsequent tracking to find results for all swept_parameters is then much faster than the initial solving. If method=:total_degree, each parameter point is solved separately by tracking the maximum number of paths (employs a total degree homotopy).

    This takes far longer but can be more reliable.

    • threading: If true, multithreaded support is activated. The number of available threads is set by the environment variable JULIA_NUM_THREADS.

    • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".

    • show_progress: Indicate whether a progress bar should be displayed.

    `,12)),s("p",null,[i[4]||(i[4]=n("Example: solving a simple harmonic oscillator ")),s("mjx-container",r,[(t(),e("svg",p,i[0]||(i[0]=[a('',1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"m"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"x"),s("mo",null,"¨")])]),s("mo",null,"+"),s("mi",null,"γ"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"x"),s("mo",null,"˙")])]),s("mo",null,"+"),s("msubsup",null,[s("mi",null,"ω"),s("mn",null,"0"),s("mn",null,"2")]),s("mi",null,"x"),s("mo",null,"="),s("mi",null,"F"),s("mi",null,"cos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"ω"),s("mi",null,"t"),s("mo",{stretchy:"false"},")")])],-1))]),i[5]||(i[5]=n(" to obtain the response as a function of ")),s("mjx-container",d,[(t(),e("svg",k,i[2]||(i[2]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),i[3]||(i[3]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω")])],-1))])]),i[7]||(i[7]=a(`
    julia
    # having obtained a Problem object, let's find steady states
+julia> range = ParameterRange(ω => LinRange(0.8,1.2,100) ) # 100 parameter sets to solve
+julia> fixed = ParameterList(m => 1, γ => 0.01, F => 0.5, ω_0 => 1)
+julia> get_steady_states(problem, range, fixed)
+
+A steady state result for 100 parameter points
+
+    Solution branches:   1
+       of which real:    1
+       of which stable:  1
+
+    Classes: stable, physical, Hopf, binary_labels

    It is also possible to create multi-dimensional solutions plots.

    julia
    # The swept parameters take precedence over fixed -> use the same fixed
+julia> range = ParameterRange(ω => LinRange(0.8,1.2,100), F => LinRange(0.1,1.0,10) ) # 100x10 parameter sets
+
+# The swept parameters take precedence over fixed -> the F in fixed is now ignored
+julia> get_steady_states(problem, range, fixed)
+
+A steady state result for 1000 parameter points
+
+    Solution branches:   1
+       of which real:    1
+       of which stable:  1
+
+    Classes: stable, physical, Hopf, binary_labels

    source

    `,4))]),i[9]||(i[9]=a(`
    # HarmonicBalance.Result — Type.
    julia
    mutable struct Result

    Stores the steady states of a HarmonicEquation.

    Fields

    • solutions::Array{Vector{Vector{ComplexF64}}}: The variable values of steady-state solutions.

    • swept_parameters::OrderedCollections.OrderedDict{Num, Vector{Union{Float64, ComplexF64}}}: Values of all parameters for all solutions.

    • fixed_parameters::OrderedCollections.OrderedDict{Num, Float64}: The parameters fixed throughout the solutions.

    • problem::Problem: The Problem used to generate this.

    • classes::Dict{String, Array}: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method classify_solutions!).

    • jacobian::Function: The Jacobian with fixed_parameters already substituted. Accepts a dictionary specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function. If problem.jacobian was false, this holds a function that rearranges the equations to find J only after numerical values are inserted (preferable in cases where the symbolic J would be very large).

    • seed::Union{Nothing, UInt32}: Seed used for the solver

    source


    Classifying solutions ​

    The solutions in Result are accompanied by similarly-sized boolean arrays stored in the dictionary Result.classes. The classes can be used by the plotting functions to show/hide/label certain solutions.

    By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with classify_solutions!.

    # HarmonicBalance.classify_solutions! — Function.
    julia
    classify_solutions!(
+    res::Result,
+    func::Union{Function, String},
+    name::String;
+    physical
+) -> Any

    Creates a solution class in res using the inequality condition (parsed into Symbolics.jl input).

    The new class is labelled with name and stored under res.classes[name].

    By default, only physical (=real) solutions are classified, false is returned for the rest.

    Example

    julia
    # solve a previously-defined problem
+res = get_steady_states(problem, swept_parameters, fixed_parameters)
+
+# classify, store in result.classes["large_amplitude"]
+classify_solutions!(res, "sqrt(u1^2 + v1^2) > 1.0" , "large_amplitude")

    source


    Sorting solutions ​

    Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.

    For stable states, the branches describe a system's behaviour under adiabatic parameter changes.

    Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.

    The function sort_solutions goes over the the raw output of get_steady_states and sorts each entry such that neighboring solution sets minimize Euclidean distance.

    Currently, sort_solutions is compatible with 1D and 2D arrays of solution sets.

    # HarmonicBalance.sort_solutions — Function.
    julia
    sort_solutions(
+    solutions::Array;
+    sorting,
+    show_progress
+) -> Any

    Sorts solutions into branches according to the method sorting.

    solutions is an n-dimensional array of Vector{Vector}. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.

    Keyword arguments

    • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".

    • show_progress: Indicate whether a progress bar should be displayed.

    source


    `,16))])}const b=l(o,[["render",c]]);export{T as __pageData,b as default}; diff --git a/v0.10.7/assets/manual_time_dependent.md.l1Mqi_ze.js b/v0.10.7/assets/manual_time_dependent.md.l1Mqi_ze.js new file mode 100644 index 00000000..2783c785 --- /dev/null +++ b/v0.10.7/assets/manual_time_dependent.md.l1Mqi_ze.js @@ -0,0 +1,29 @@ +import{_ as i,c as a,a4 as e,o as n}from"./chunks/framework.D3a-l5sU.js";const E=JSON.parse('{"title":"Time evolution","description":"","frontmatter":{},"headers":[],"relativePath":"manual/time_dependent.md","filePath":"manual/time_dependent.md"}'),t={name:"manual/time_dependent.md"};function l(p,s,h,k,r,d){return n(),a("div",null,s[0]||(s[0]=[e(`

    Time evolution ​

    Generally, solving the ODE of oscillatory systems in time requires numerically tracking the oscillations. This is a computationally expensive process; however, using the harmonic ansatz removes the oscillatory time-dependence. Simulating instead the harmonic variables of a HarmonicEquation is vastly more efficient - a steady state of the system appears as a fixed point in multidimensional space rather than an oscillatory function.

    The Extention TimeEvolution is used to interface HarmonicEquation with the solvers contained in OrdinaryDiffEq.jl. Time-dependent parameter sweeps are defined using the object ParameterSweep. To use the TimeEvolution extension, one must first load the OrdinaryDiffEq.jl package.

    # SciMLBase.ODEProblem — Method.
    julia
    ODEProblem(
+        eom::HarmonicEquation;
+        fixed_parameters,
+        x0::Vector,
+        sweep::ParameterSweep,
+        timespan::Tuple
+        )

    Creates an ODEProblem object used by OrdinaryDiffEq.jl from the equations in eom to simulate time-evolution within timespan. fixed_parameters must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If x0 is specified, it is used as an initial condition; otherwise the values from fixed_parameters are used.

    source


    # HarmonicBalance.ParameterSweep — Type.

    Represents a sweep of one or more parameters of a HarmonicEquation. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.

    Sweeps of different variables can be combined using +.

    Fields

    • functions::Dict{Num, Function}: Maps each swept parameter to a function.

    Examples

    julia
    # create a sweep of parameter a from 0 to 1 over time 0 -> 100
+julia> @variables a,b;
+julia> sweep = ParameterSweep(a => [0., 1.], (0, 100));
+julia> sweep[a](50)
+0.5
+julia> sweep[a](200)
+1.0
+
+# do the same, varying two parameters simultaneously
+julia> sweep = ParameterSweep([a => [0.,1.], b => [0., 1.]], (0,100))

    Successive sweeps can be combined,

    julia
    sweep1 = ParameterSweep(ω => [0.95, 1.0], (0, 2e4))
+sweep2 = ParameterSweep(λ => [0.05, 0.01], (2e4, 4e4))
+sweep = sweep1 + sweep2

    multiple parameters can be swept simultaneously,

    julia
    sweep = ParameterSweep([ω => [0.95;1.0], λ => [5e-2;1e-2]], (0, 2e4))

    and custom sweep functions may be used.

    julia
    ωfunc(t) = cos(t)
+sweep = ParameterSweep(ω => ωfunc)

    source


    Plotting ​

    # RecipesBase.plot — Method.
    julia
    plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

    Plot a function f of a time-dependent solution soln of harm_eq.

    As a function of time

    plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

    f is parsed by Symbolics.jl

    parametric plots

    plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)

    Parametric plot of f[1] against f[2]

    Also callable as plot!

    source


    Miscellaneous ​

    Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.

    # HarmonicBalance.is_stable — Function.
    julia
    is_stable(
+    soln::OrderedCollections.OrderedDict{Num, ComplexF64},
+    eom::HarmonicEquation;
+    timespan,
+    tol,
+    perturb_initial
+)

    Numerically investigate the stability of a solution soln of eom within timespan. The initial condition is displaced by perturb_initial.

    Return true the solution evolves within tol of the initial value (interpreted as stable).

    source

    julia
    is_stable(
+    soln::OrderedCollections.OrderedDict{Num, ComplexF64},
+    res::Result;
+    kwargs...
+) -> Any

    Returns true if the solution soln of the Result res is stable. Stable solutions are real and have all Jacobian eigenvalues Re[λ] <= 0. im_tol : an absolute threshold to distinguish real/complex numbers. rel_tol: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)

    source


    `,14)]))}const c=i(t,[["render",l]]);export{E as __pageData,c as default}; diff --git a/v0.10.7/assets/manual_time_dependent.md.l1Mqi_ze.lean.js b/v0.10.7/assets/manual_time_dependent.md.l1Mqi_ze.lean.js new file mode 100644 index 00000000..2783c785 --- /dev/null +++ b/v0.10.7/assets/manual_time_dependent.md.l1Mqi_ze.lean.js @@ -0,0 +1,29 @@ +import{_ as i,c as a,a4 as e,o as n}from"./chunks/framework.D3a-l5sU.js";const E=JSON.parse('{"title":"Time evolution","description":"","frontmatter":{},"headers":[],"relativePath":"manual/time_dependent.md","filePath":"manual/time_dependent.md"}'),t={name:"manual/time_dependent.md"};function l(p,s,h,k,r,d){return n(),a("div",null,s[0]||(s[0]=[e(`

    Time evolution ​

    Generally, solving the ODE of oscillatory systems in time requires numerically tracking the oscillations. This is a computationally expensive process; however, using the harmonic ansatz removes the oscillatory time-dependence. Simulating instead the harmonic variables of a HarmonicEquation is vastly more efficient - a steady state of the system appears as a fixed point in multidimensional space rather than an oscillatory function.

    The Extention TimeEvolution is used to interface HarmonicEquation with the solvers contained in OrdinaryDiffEq.jl. Time-dependent parameter sweeps are defined using the object ParameterSweep. To use the TimeEvolution extension, one must first load the OrdinaryDiffEq.jl package.

    # SciMLBase.ODEProblem — Method.
    julia
    ODEProblem(
+        eom::HarmonicEquation;
+        fixed_parameters,
+        x0::Vector,
+        sweep::ParameterSweep,
+        timespan::Tuple
+        )

    Creates an ODEProblem object used by OrdinaryDiffEq.jl from the equations in eom to simulate time-evolution within timespan. fixed_parameters must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If x0 is specified, it is used as an initial condition; otherwise the values from fixed_parameters are used.

    source


    # HarmonicBalance.ParameterSweep — Type.

    Represents a sweep of one or more parameters of a HarmonicEquation. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.

    Sweeps of different variables can be combined using +.

    Fields

    • functions::Dict{Num, Function}: Maps each swept parameter to a function.

    Examples

    julia
    # create a sweep of parameter a from 0 to 1 over time 0 -> 100
+julia> @variables a,b;
+julia> sweep = ParameterSweep(a => [0., 1.], (0, 100));
+julia> sweep[a](50)
+0.5
+julia> sweep[a](200)
+1.0
+
+# do the same, varying two parameters simultaneously
+julia> sweep = ParameterSweep([a => [0.,1.], b => [0., 1.]], (0,100))

    Successive sweeps can be combined,

    julia
    sweep1 = ParameterSweep(ω => [0.95, 1.0], (0, 2e4))
+sweep2 = ParameterSweep(λ => [0.05, 0.01], (2e4, 4e4))
+sweep = sweep1 + sweep2

    multiple parameters can be swept simultaneously,

    julia
    sweep = ParameterSweep([ω => [0.95;1.0], λ => [5e-2;1e-2]], (0, 2e4))

    and custom sweep functions may be used.

    julia
    ωfunc(t) = cos(t)
+sweep = ParameterSweep(ω => ωfunc)

    source


    Plotting ​

    # RecipesBase.plot — Method.
    julia
    plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

    Plot a function f of a time-dependent solution soln of harm_eq.

    As a function of time

    plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

    f is parsed by Symbolics.jl

    parametric plots

    plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)

    Parametric plot of f[1] against f[2]

    Also callable as plot!

    source


    Miscellaneous ​

    Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.

    # HarmonicBalance.is_stable — Function.
    julia
    is_stable(
+    soln::OrderedCollections.OrderedDict{Num, ComplexF64},
+    eom::HarmonicEquation;
+    timespan,
+    tol,
+    perturb_initial
+)

    Numerically investigate the stability of a solution soln of eom within timespan. The initial condition is displaced by perturb_initial.

    Return true the solution evolves within tol of the initial value (interpreted as stable).

    source

    julia
    is_stable(
+    soln::OrderedCollections.OrderedDict{Num, ComplexF64},
+    res::Result;
+    kwargs...
+) -> Any

    Returns true if the solution soln of the Result res is stable. Stable solutions are real and have all Jacobian eigenvalues Re[λ] <= 0. im_tol : an absolute threshold to distinguish real/complex numbers. rel_tol: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)

    source


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    Classifying solutions ​

    Given that you obtained some steady states for a parameter sweep of a specific model it can be useful to classify these solution. Let us consider a simple pametric oscillator

    julia
    using HarmonicBalance
+
+@variables ω₀ γ λ α ω t x(t)
+
+natural_equation = d(d(x, t), t) + γ * d(x, t) + (ω₀^2 - λ * cos(2 * ω * t)) * x + α * x^3
+diff_eq = DifferentialEquation(natural_equation, x)
+
+add_harmonic!(diff_eq, x, ω);
+
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω₀, λ
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+-(1//2)*u1(T)*λ + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω₀^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ 0
+
+Differential(T)(v1(T))*γ + (1//2)*v1(T)*λ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω₀^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0
    `,4)),i("p",null,[s[4]||(s[4]=a("We performe a 2d sweep in the driving frequency ")),i("mjx-container",o,[(e(),t("svg",d,s[0]||(s[0]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[1]||(s[1]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω")])],-1))]),s[5]||(s[5]=a(" and driving strength ")),i("mjx-container",g,[(e(),t("svg",c,s[2]||(s[2]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),s[3]||(s[3]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"λ")])],-1))]),s[6]||(s[6]=a(":"))]),s[12]||(s[12]=n(`
    julia
    fixed = (ω₀ => 1.0, γ => 0.002, α => 1.0)
+varied = (ω => range(0.99, 1.01, 100), λ => range(1e-6, 0.03, 100))
+
+result_2D = get_steady_states(harmonic_eq, varied, fixed, threading=true)
    A steady state result for 10000 parameter points
+
+Solution branches:   5
+   of which real:    5
+   of which stable:  5
+
+Classes: stable, physical, Hopf, binary_labels

    By default the steady states of the system are classified by four different catogaries:

    • physical: Solutions that are physical, i.e., all variables are purely real.

    • stable: Solutions that are stable, i.e., all eigenvalues of the Jacobian have negative real parts.

    • Hopf: Solutions that are physical and have exactly two Jacobian eigenvalues with positive real parts, which are complex conjugates of each other. The class can help to identify regions where a limit cycle is present due to a Hopf bifurcation. See also the tutorial on limit cycles.

    • binary_labels: each region in the parameter sweep receives an identifier based on its permutation of stable branches. This allows to distinguish between different phases, which may have the same number of stable solutions.

    We can plot the number of stable solutions, giving the phase diagram

    julia
    plot_phase_diagram(result_2D, class="stable")

    ',7)),i("p",null,[s[9]||(s[9]=a("If we plot the a cut at ")),i("mjx-container",E,[(e(),t("svg",u,s[7]||(s[7]=[n('',1)]))),s[8]||(s[8]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"λ"),i("mo",null,"="),i("mn",null,"0.01")])],-1))]),s[10]||(s[10]=a(", we see that in the blue region only one stable solution exists with zero amplitude:"))]),s[13]||(s[13]=n(`
    julia
    plot(result_2D, y="√(u1^2+v1^2)", cut=λ => 0.01, class="stable") |> display
    julia
    get_single_solution(result_2D; branch=1, index=(1, 1))
    OrderedCollections.OrderedDict{Num, ComplexF64} with 7 entries:
+  u1 => -3.35208e-249-5.36333e-248im
+  v1 => -7.59806e-248+1.45257e-248im
+  ω  => 0.99+0.0im
+  λ  => 1.0e-6+0.0im
+  ω₀ => 1.0+0.0im
+  γ  => 0.002+0.0im
+  α  => 1.0+0.0im

    This solution becomes stable again outside the green lobe. Also called Mathieu lobe. Indeed, we can classify the zero amplitude solution by adding an extra catagory as a class:

    julia
    classify_solutions!(result_2D, "sqrt(u1^2 + v1^2) < 0.001", "zero")
+result_2D
    A steady state result for 10000 parameter points
+
+Solution branches:   5
+   of which real:    5
+   of which stable:  5
+
+Classes: zero, stable, physical, Hopf, binary_labels

    We can visualize the zero amplitude solution:

    julia
    plot_phase_diagram(result_2D, class=["zero", "stable"])

    This shows that inside the Mathieu lobe the zero amplitude solution becomes unstable due to the parametric drive being resonant with the oscillator.

    We can also visualize the equi-amplitude curves of the solutions:

    julia
    classify_solutions!(result_2D, "sqrt(u1^2 + v1^2) > 0.12", "large amplitude")
+plot_phase_diagram(result_2D, class=["large amplitude", "stable"])

    ',13))])}const x=l(r,[["render",y]]);export{T as __pageData,x as default}; diff --git a/v0.10.7/assets/tutorials_classification.md.no29juWd.lean.js b/v0.10.7/assets/tutorials_classification.md.no29juWd.lean.js new file mode 100644 index 00000000..eaae7e04 --- /dev/null +++ b/v0.10.7/assets/tutorials_classification.md.no29juWd.lean.js @@ -0,0 +1,45 @@ +import{_ as l,c as t,a4 as n,j as i,a,o as e}from"./chunks/framework.D3a-l5sU.js";const p="/HarmonicBalance.jl/v0.10.7/assets/ohndoqr.DDG3oKdt.png",h="/HarmonicBalance.jl/v0.10.7/assets/qdsxmjc.Nsw0w518.png",k="/HarmonicBalance.jl/v0.10.7/assets/rgizkmj.CHo32oEM.png",T=JSON.parse('{"title":"Classifying solutions","description":"","frontmatter":{},"headers":[],"relativePath":"tutorials/classification.md","filePath":"tutorials/classification.md"}'),r={name:"tutorials/classification.md"},o={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},d={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"},E={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.359ex",height:"1.756ex",role:"img",focusable:"false",viewBox:"0 -694 3694.6 776","aria-hidden":"true"};function y(m,s,F,b,v,C){return e(),t("div",null,[s[11]||(s[11]=n(`

    Classifying solutions ​

    Given that you obtained some steady states for a parameter sweep of a specific model it can be useful to classify these solution. Let us consider a simple pametric oscillator

    julia
    using HarmonicBalance
+
+@variables ω₀ γ λ α ω t x(t)
+
+natural_equation = d(d(x, t), t) + γ * d(x, t) + (ω₀^2 - λ * cos(2 * ω * t)) * x + α * x^3
+diff_eq = DifferentialEquation(natural_equation, x)
+
+add_harmonic!(diff_eq, x, ω);
+
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω₀, λ
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+-(1//2)*u1(T)*λ + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω₀^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ 0
+
+Differential(T)(v1(T))*γ + (1//2)*v1(T)*λ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω₀^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0
    `,4)),i("p",null,[s[4]||(s[4]=a("We performe a 2d sweep in the driving frequency ")),i("mjx-container",o,[(e(),t("svg",d,s[0]||(s[0]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[1]||(s[1]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω")])],-1))]),s[5]||(s[5]=a(" and driving strength ")),i("mjx-container",g,[(e(),t("svg",c,s[2]||(s[2]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),s[3]||(s[3]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"λ")])],-1))]),s[6]||(s[6]=a(":"))]),s[12]||(s[12]=n(`
    julia
    fixed = (ω₀ => 1.0, γ => 0.002, α => 1.0)
+varied = (ω => range(0.99, 1.01, 100), λ => range(1e-6, 0.03, 100))
+
+result_2D = get_steady_states(harmonic_eq, varied, fixed, threading=true)
    A steady state result for 10000 parameter points
+
+Solution branches:   5
+   of which real:    5
+   of which stable:  5
+
+Classes: stable, physical, Hopf, binary_labels

    By default the steady states of the system are classified by four different catogaries:

    • physical: Solutions that are physical, i.e., all variables are purely real.

    • stable: Solutions that are stable, i.e., all eigenvalues of the Jacobian have negative real parts.

    • Hopf: Solutions that are physical and have exactly two Jacobian eigenvalues with positive real parts, which are complex conjugates of each other. The class can help to identify regions where a limit cycle is present due to a Hopf bifurcation. See also the tutorial on limit cycles.

    • binary_labels: each region in the parameter sweep receives an identifier based on its permutation of stable branches. This allows to distinguish between different phases, which may have the same number of stable solutions.

    We can plot the number of stable solutions, giving the phase diagram

    julia
    plot_phase_diagram(result_2D, class="stable")

    ',7)),i("p",null,[s[9]||(s[9]=a("If we plot the a cut at ")),i("mjx-container",E,[(e(),t("svg",u,s[7]||(s[7]=[n('',1)]))),s[8]||(s[8]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"λ"),i("mo",null,"="),i("mn",null,"0.01")])],-1))]),s[10]||(s[10]=a(", we see that in the blue region only one stable solution exists with zero amplitude:"))]),s[13]||(s[13]=n(`
    julia
    plot(result_2D, y="√(u1^2+v1^2)", cut=λ => 0.01, class="stable") |> display
    julia
    get_single_solution(result_2D; branch=1, index=(1, 1))
    OrderedCollections.OrderedDict{Num, ComplexF64} with 7 entries:
+  u1 => -3.35208e-249-5.36333e-248im
+  v1 => -7.59806e-248+1.45257e-248im
+  ω  => 0.99+0.0im
+  λ  => 1.0e-6+0.0im
+  ω₀ => 1.0+0.0im
+  γ  => 0.002+0.0im
+  α  => 1.0+0.0im

    This solution becomes stable again outside the green lobe. Also called Mathieu lobe. Indeed, we can classify the zero amplitude solution by adding an extra catagory as a class:

    julia
    classify_solutions!(result_2D, "sqrt(u1^2 + v1^2) < 0.001", "zero")
+result_2D
    A steady state result for 10000 parameter points
+
+Solution branches:   5
+   of which real:    5
+   of which stable:  5
+
+Classes: zero, stable, physical, Hopf, binary_labels

    We can visualize the zero amplitude solution:

    julia
    plot_phase_diagram(result_2D, class=["zero", "stable"])

    This shows that inside the Mathieu lobe the zero amplitude solution becomes unstable due to the parametric drive being resonant with the oscillator.

    We can also visualize the equi-amplitude curves of the solutions:

    julia
    classify_solutions!(result_2D, "sqrt(u1^2 + v1^2) > 0.12", "large amplitude")
+plot_phase_diagram(result_2D, class=["large amplitude", "stable"])

    ',13))])}const x=l(r,[["render",y]]);export{T as __pageData,x as default}; diff --git a/v0.10.7/assets/tutorials_index.md.1D0D6Jn-.js b/v0.10.7/assets/tutorials_index.md.1D0D6Jn-.js new file mode 100644 index 00000000..f0bc9460 --- /dev/null +++ b/v0.10.7/assets/tutorials_index.md.1D0D6Jn-.js @@ -0,0 +1 @@ +import{G as n}from"./chunks/Gallery.DYdk1hsn.js";import{d as o,c as i,j as t,a as s,G as r,o as l}from"./chunks/framework.D3a-l5sU.js";const u=JSON.parse('{"title":"Tutorials","description":"","frontmatter":{},"headers":[],"relativePath":"tutorials/index.md","filePath":"tutorials/index.md"}'),c={name:"tutorials/index.md"},g=o({...c,setup(p){const a=[{href:"steady_states",src:"https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/simple_Duffing/response_single.png",caption:"Steady states",desc:"How to get the steady states of the harmonic equations."},{href:"classification",src:"https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/parametron/2d_phase_diagram.png",caption:"Classifying solutions",desc:"Learn how to add different types of drives."},{href:"linear_response",src:"https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/linear_response/nonlin_F_noise.png",caption:"Linear response",desc:"Learn how to compute the linear response of a steady state."},{href:"time_dependent",src:"https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/time_dependent/evo_to_steady.png",caption:"Stroboscopic evolution",desc:"Learn how to investigate stroboscopic time evolution."},{href:"limit_cycles",src:"https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/limit_cycles/vdp_degenerate.png",caption:"Limit cycles",desc:"Learn how to find the limit cycles of your system."}];return(d,e)=>(l(),i("div",null,[e[0]||(e[0]=t("h1",{id:"tutorials",tabindex:"-1"},[s("Tutorials "),t("a",{class:"header-anchor",href:"#tutorials","aria-label":'Permalink to "Tutorials {#tutorials}"'},"​")],-1)),e[1]||(e[1]=t("p",null,[s("We show the capabilities of the package by providing a series of tutorials. Examples of other systems can be found in the "),t("a",{href:"/HarmonicBalance.jl/v0.10.7/examples/index#examples"},"examples"),s(" tab.")],-1)),r(n,{images:a})]))}});export{u as __pageData,g as default}; diff --git a/v0.10.7/assets/tutorials_index.md.1D0D6Jn-.lean.js b/v0.10.7/assets/tutorials_index.md.1D0D6Jn-.lean.js new file mode 100644 index 00000000..f0bc9460 --- /dev/null +++ b/v0.10.7/assets/tutorials_index.md.1D0D6Jn-.lean.js @@ -0,0 +1 @@ +import{G as n}from"./chunks/Gallery.DYdk1hsn.js";import{d as o,c as i,j as t,a as s,G as r,o as l}from"./chunks/framework.D3a-l5sU.js";const u=JSON.parse('{"title":"Tutorials","description":"","frontmatter":{},"headers":[],"relativePath":"tutorials/index.md","filePath":"tutorials/index.md"}'),c={name:"tutorials/index.md"},g=o({...c,setup(p){const a=[{href:"steady_states",src:"https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/simple_Duffing/response_single.png",caption:"Steady states",desc:"How to get the steady states of the harmonic equations."},{href:"classification",src:"https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/parametron/2d_phase_diagram.png",caption:"Classifying solutions",desc:"Learn how to add different types of drives."},{href:"linear_response",src:"https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/linear_response/nonlin_F_noise.png",caption:"Linear response",desc:"Learn how to compute the linear response of a steady state."},{href:"time_dependent",src:"https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/time_dependent/evo_to_steady.png",caption:"Stroboscopic evolution",desc:"Learn how to investigate stroboscopic time evolution."},{href:"limit_cycles",src:"https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/limit_cycles/vdp_degenerate.png",caption:"Limit cycles",desc:"Learn how to find the limit cycles of your system."}];return(d,e)=>(l(),i("div",null,[e[0]||(e[0]=t("h1",{id:"tutorials",tabindex:"-1"},[s("Tutorials "),t("a",{class:"header-anchor",href:"#tutorials","aria-label":'Permalink to "Tutorials {#tutorials}"'},"​")],-1)),e[1]||(e[1]=t("p",null,[s("We show the capabilities of the package by providing a series of tutorials. 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    Limit cycles ​

    In contrast to the previous tutorials, limit cycle problems feature harmonic(s) whose numerical value is not imposed externally. We shall construct our HarmonicEquation as usual, but identify this harmonic as an extra variable, rather than a fixed parameter.

    Non-driven system - the van der Pol oscillator ​

    Here we solve the equation of motion of the van der Pol oscillator. This is a single-variable second-order ODE with continuous time-translation symmetry (i.e., no 'clock' imposing a frequency and/or phase), which displays periodic solutions known as relaxation oscillations. For more detail, refer also to arXiv:2308.06092.

    julia
    using HarmonicBalance
+@variables ω_lc, t, ω0, x(t), μ
+diff_eq = DifferentialEquation(d(d(x,t),t) - μ*(1-x^2) * d(x,t) + x, x)
    System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ;   
+
+x(t) + Differential(t)(Differential(t)(x(t))) - (1 - (x(t)^2))*Differential(t)(x(t))*μ ~ 0
    `,6)),t("p",null,[s[8]||(s[8]=a("Choosing to expand the motion of ")),t("mjx-container",h,[(l(),e("svg",m,s[0]||(s[0]=[i('',1)]))),s[1]||(s[1]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),s[9]||(s[9]=a(" using ")),t("mjx-container",k,[(l(),e("svg",u,s[2]||(s[2]=[i('',1)]))),s[3]||(s[3]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"l"),t("mi",null,"c")])])])],-1))]),s[10]||(s[10]=a(", ")),t("mjx-container",g,[(l(),e("svg",c,s[4]||(s[4]=[i('',1)]))),s[5]||(s[5]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"l"),t("mi",null,"c")])])])],-1))]),s[11]||(s[11]=a(" and ")),t("mjx-container",f,[(l(),e("svg",y,s[6]||(s[6]=[i('',1)]))),s[7]||(s[7]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"5"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"l"),t("mi",null,"c")])])])],-1))]),s[12]||(s[12]=a(", we define"))]),s[104]||(s[104]=i(`
    julia
    foreach(1:2:5) do i
+  add_harmonic!(diff_eq, x, i*ω_lc)
+end;

    and obtain 6 harmonic equations,

    julia
    harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 6 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T), u3(T), v3(T)
+Parameters: ω_lc, μ
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ω_lct) + v1(T)*sin(ω_lct) + u2(T)*cos(3ω_lct) + v2(T)*sin(3ω_lct) + u3(T)*cos(5ω_lct) + v3(T)*sin(5ω_lct)
+
+Harmonic equations:
+
+u1(T) + (2//1)*Differential(T)(v1(T))*ω_lc - Differential(T)(u1(T))*μ - u1(T)*(ω_lc^2) - v1(T)*μ*ω_lc + (1//4)*(u1(T)^2)*Differential(T)(u2(T))*μ + (3//4)*(u1(T)^2)*Differential(T)(u1(T))*μ + u1(T)*Differential(T)(u3(T))*u3(T)*μ + (1//2)*u1(T)*Differential(T)(u3(T))*u2(T)*μ + u1(T)*Differential(T)(v3(T))*v3(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*v2(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*Differential(T)(v1(T))*v2(T)*μ + (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*μ + u1(T)*v2(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*Differential(T)(u2(T))*u3(T)*μ + u1(T)*Differential(T)(u2(T))*u2(T)*μ + (1//2)*u1(T)*Differential(T)(u1(T))*u2(T)*μ - (1//4)*Differential(T)(u3(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(u3(T))*v2(T)*v1(T)*μ + (1//4)*Differential(T)(u3(T))*(u2(T)^2)*μ + (1//2)*Differential(T)(v3(T))*v2(T)*u2(T)*μ + (1//2)*Differential(T)(v3(T))*v1(T)*u2(T)*μ + (1//2)*(v3(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v3(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*v3(T)*v2(T)*Differential(T)(u2(T))*μ + (1//2)*v3(T)*v2(T)*Differential(T)(u1(T))*μ + (1//2)*v3(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*v3(T)*u2(T)*Differential(T)(v2(T))*μ - (1//2)*Differential(T)(v1(T))*v2(T)*u3(T)*μ - (1//2)*Differential(T)(v1(T))*v1(T)*u2(T)*μ + (1//2)*(v2(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v2(T)*v1(T)*Differential(T)(u1(T))*μ - (1//2)*v2(T)*u3(T)*Differential(T)(v2(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(u2(T))*μ + (1//4)*(v1(T)^2)*Differential(T)(u1(T))*μ - (1//2)*v1(T)*u3(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u2(T))*u3(T)*u2(T)*μ + (1//2)*(u3(T)^2)*Differential(T)(u1(T))*μ + (1//2)*u3(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*Differential(T)(u1(T))*(u2(T)^2)*μ + (1//4)*(u1(T)^2)*v2(T)*μ*ω_lc + (1//4)*(u1(T)^2)*v1(T)*μ*ω_lc + (1//2)*u1(T)*v3(T)*u2(T)*μ*ω_lc - (1//2)*u1(T)*v2(T)*u3(T)*μ*ω_lc - (1//2)*u1(T)*v1(T)*u2(T)*μ*ω_lc + (1//2)*(v3(T)^2)*v1(T)*μ*ω_lc + (1//4)*v3(T)*(v2(T)^2)*μ*ω_lc - (1//2)*v3(T)*v2(T)*v1(T)*μ*ω_lc - (1//4)*v3(T)*(u2(T)^2)*μ*ω_lc + (1//2)*(v2(T)^2)*v1(T)*μ*ω_lc - (1//4)*v2(T)*(v1(T)^2)*μ*ω_lc + (1//2)*v2(T)*u3(T)*u2(T)*μ*ω_lc + (1//4)*(v1(T)^3)*μ*ω_lc + (1//2)*v1(T)*(u3(T)^2)*μ*ω_lc - (1//2)*v1(T)*u3(T)*u2(T)*μ*ω_lc + (1//2)*v1(T)*(u2(T)^2)*μ*ω_lc ~ 0
+
+v1(T) - Differential(T)(v1(T))*μ - (2//1)*Differential(T)(u1(T))*ω_lc + u1(T)*μ*ω_lc - v1(T)*(ω_lc^2) + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*Differential(T)(u3(T))*v2(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*u2(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(u2(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*u1(T)*v2(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*u3(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u3(T))*v2(T)*u2(T)*μ + Differential(T)(u3(T))*v1(T)*u3(T)*μ - (1//2)*Differential(T)(u3(T))*v1(T)*u2(T)*μ + Differential(T)(v3(T))*v3(T)*v1(T)*μ + (1//4)*Differential(T)(v3(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(v3(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v3(T))*(u2(T)^2)*μ + (1//2)*(v3(T)^2)*Differential(T)(v1(T))*μ - (1//2)*v3(T)*Differential(T)(v1(T))*v2(T)*μ + (1//2)*v3(T)*v2(T)*Differential(T)(v2(T))*μ - (1//2)*v3(T)*v1(T)*Differential(T)(v2(T))*μ - (1//2)*v3(T)*Differential(T)(u2(T))*u2(T)*μ + (1//2)*v3(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*Differential(T)(v1(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(v1(T))*v2(T)*v1(T)*μ + (3//4)*Differential(T)(v1(T))*(v1(T)^2)*μ + (1//2)*Differential(T)(v1(T))*(u3(T)^2)*μ - (1//2)*Differential(T)(v1(T))*u3(T)*u2(T)*μ + (1//2)*Differential(T)(v1(T))*(u2(T)^2)*μ + v2(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*v2(T)*Differential(T)(u2(T))*u3(T)*μ - (1//2)*v2(T)*u3(T)*Differential(T)(u1(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*v1(T)*Differential(T)(u2(T))*u3(T)*μ + v1(T)*Differential(T)(u2(T))*u2(T)*μ - (1//2)*v1(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*u3(T)*u2(T)*Differential(T)(v2(T))*μ - (1//4)*(u1(T)^3)*μ*ω_lc - (1//4)*(u1(T)^2)*u2(T)*μ*ω_lc - (1//2)*u1(T)*(v3(T)^2)*μ*ω_lc - (1//2)*u1(T)*v3(T)*v2(T)*μ*ω_lc - (1//2)*u1(T)*(v2(T)^2)*μ*ω_lc - (1//2)*u1(T)*v2(T)*v1(T)*μ*ω_lc - (1//4)*u1(T)*(v1(T)^2)*μ*ω_lc - (1//2)*u1(T)*(u3(T)^2)*μ*ω_lc - (1//2)*u1(T)*u3(T)*u2(T)*μ*ω_lc - (1//2)*u1(T)*(u2(T)^2)*μ*ω_lc - (1//2)*v3(T)*v2(T)*u2(T)*μ*ω_lc - (1//2)*v3(T)*v1(T)*u2(T)*μ*ω_lc + (1//4)*(v2(T)^2)*u3(T)*μ*ω_lc + (1//2)*v2(T)*v1(T)*u3(T)*μ*ω_lc + (1//4)*(v1(T)^2)*u2(T)*μ*ω_lc - (1//4)*u3(T)*(u2(T)^2)*μ*ω_lc ~ 0
+
+u2(T) - Differential(T)(u2(T))*μ + (6//1)*Differential(T)(v2(T))*ω_lc - (3//1)*v2(T)*μ*ω_lc - (9//1)*u2(T)*(ω_lc^2) + (1//4)*(u1(T)^2)*Differential(T)(u3(T))*μ + (1//2)*(u1(T)^2)*Differential(T)(u2(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*Differential(T)(u3(T))*u2(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*v2(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*v1(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(v1(T))*μ + (1//2)*u1(T)*v3(T)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*μ + (1//2)*u1(T)*Differential(T)(u2(T))*u3(T)*μ + (1//2)*u1(T)*u3(T)*Differential(T)(u1(T))*μ + u1(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*Differential(T)(u3(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(u3(T))*(v1(T)^2)*μ + Differential(T)(u3(T))*u3(T)*u2(T)*μ + Differential(T)(v3(T))*v3(T)*u2(T)*μ - (1//2)*Differential(T)(v3(T))*v1(T)*u2(T)*μ + (1//2)*(v3(T)^2)*Differential(T)(u2(T))*μ - (1//2)*v3(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*v3(T)*v2(T)*Differential(T)(u1(T))*μ - (1//2)*v3(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*v3(T)*v1(T)*Differential(T)(u1(T))*μ + (1//2)*Differential(T)(v1(T))*v2(T)*u3(T)*μ - (1//2)*Differential(T)(v1(T))*v1(T)*u3(T)*μ + Differential(T)(v1(T))*v1(T)*u2(T)*μ + (1//4)*(v2(T)^2)*Differential(T)(u2(T))*μ + (1//2)*v2(T)*u2(T)*Differential(T)(v2(T))*μ + (1//2)*(v1(T)^2)*Differential(T)(u2(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v1(T)*u3(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u2(T))*(u3(T)^2)*μ + (3//4)*Differential(T)(u2(T))*(u2(T)^2)*μ + (1//2)*u3(T)*Differential(T)(u1(T))*u2(T)*μ + (3//4)*(u1(T)^2)*v3(T)*μ*ω_lc + (3//2)*(u1(T)^2)*v2(T)*μ*ω_lc + (3//4)*(u1(T)^2)*v1(T)*μ*ω_lc + (3//2)*u1(T)*v3(T)*u2(T)*μ*ω_lc - (3//2)*u1(T)*v2(T)*u3(T)*μ*ω_lc - (3//2)*u1(T)*v1(T)*u3(T)*μ*ω_lc + (3//2)*(v3(T)^2)*v2(T)*μ*ω_lc + (3//2)*v3(T)*v2(T)*v1(T)*μ*ω_lc - (3//4)*v3(T)*(v1(T)^2)*μ*ω_lc + (3//4)*(v2(T)^3)*μ*ω_lc + (3//2)*v2(T)*(v1(T)^2)*μ*ω_lc + (3//2)*v2(T)*(u3(T)^2)*μ*ω_lc + (3//4)*v2(T)*(u2(T)^2)*μ*ω_lc - (1//4)*(v1(T)^3)*μ*ω_lc + (3//2)*v1(T)*u3(T)*u2(T)*μ*ω_lc ~ 0
+
+v2(T) - (6//1)*Differential(T)(u2(T))*ω_lc - Differential(T)(v2(T))*μ - (9//1)*v2(T)*(ω_lc^2) + (3//1)*u2(T)*μ*ω_lc + (1//4)*(u1(T)^2)*Differential(T)(v3(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*μ + (1//2)*(u1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*Differential(T)(u3(T))*v2(T)*μ - (1//2)*u1(T)*Differential(T)(u3(T))*v1(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*u2(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*v3(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*u3(T)*μ + u1(T)*v2(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*u3(T)*Differential(T)(v2(T))*μ + Differential(T)(u3(T))*v2(T)*u3(T)*μ + (1//2)*Differential(T)(u3(T))*v1(T)*u2(T)*μ + Differential(T)(v3(T))*v3(T)*v2(T)*μ + (1//2)*Differential(T)(v3(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v3(T))*(v1(T)^2)*μ + (1//2)*(v3(T)^2)*Differential(T)(v2(T))*μ + (1//2)*v3(T)*Differential(T)(v1(T))*v2(T)*μ - (1//2)*v3(T)*Differential(T)(v1(T))*v1(T)*μ + (1//2)*v3(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*v3(T)*Differential(T)(u1(T))*u2(T)*μ + Differential(T)(v1(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v1(T))*(v1(T)^2)*μ + (1//2)*Differential(T)(v1(T))*u3(T)*u2(T)*μ + (3//4)*(v2(T)^2)*Differential(T)(v2(T))*μ + (1//2)*v2(T)*Differential(T)(u2(T))*u2(T)*μ - (1//2)*v2(T)*u3(T)*Differential(T)(u1(T))*μ + (1//2)*(v1(T)^2)*Differential(T)(v2(T))*μ + (1//2)*v1(T)*Differential(T)(u2(T))*u3(T)*μ - (1//2)*v1(T)*u3(T)*Differential(T)(u1(T))*μ + (1//2)*(u3(T)^2)*Differential(T)(v2(T))*μ + (1//4)*(u2(T)^2)*Differential(T)(v2(T))*μ - (1//4)*(u1(T)^3)*μ*ω_lc - (3//4)*(u1(T)^2)*u3(T)*μ*ω_lc - (3//2)*(u1(T)^2)*u2(T)*μ*ω_lc - (3//2)*u1(T)*v3(T)*v2(T)*μ*ω_lc - (3//2)*u1(T)*v3(T)*v1(T)*μ*ω_lc + (3//4)*u1(T)*(v1(T)^2)*μ*ω_lc - (3//2)*u1(T)*u3(T)*u2(T)*μ*ω_lc - (3//2)*(v3(T)^2)*u2(T)*μ*ω_lc + (3//2)*v3(T)*v1(T)*u2(T)*μ*ω_lc - (3//4)*(v2(T)^2)*u2(T)*μ*ω_lc - (3//2)*v2(T)*v1(T)*u3(T)*μ*ω_lc + (3//4)*(v1(T)^2)*u3(T)*μ*ω_lc - (3//2)*(v1(T)^2)*u2(T)*μ*ω_lc - (3//2)*(u3(T)^2)*u2(T)*μ*ω_lc - (3//4)*(u2(T)^3)*μ*ω_lc ~ 0
+
+u3(T) - Differential(T)(u3(T))*μ + (10//1)*Differential(T)(v3(T))*ω_lc - (5//1)*v3(T)*μ*ω_lc - (25//1)*u3(T)*(ω_lc^2) + (1//2)*(u1(T)^2)*Differential(T)(u3(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(u2(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*v2(T)*μ - (1//2)*u1(T)*v2(T)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*Differential(T)(u2(T))*u2(T)*μ + u1(T)*u3(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*Differential(T)(u1(T))*u2(T)*μ + (1//4)*Differential(T)(u3(T))*(v3(T)^2)*μ + (1//2)*Differential(T)(u3(T))*(v2(T)^2)*μ + (1//2)*Differential(T)(u3(T))*(v1(T)^2)*μ + (3//4)*Differential(T)(u3(T))*(u3(T)^2)*μ + (1//2)*Differential(T)(u3(T))*(u2(T)^2)*μ + (1//2)*Differential(T)(v3(T))*v3(T)*u3(T)*μ + (1//2)*Differential(T)(v1(T))*v2(T)*u2(T)*μ + Differential(T)(v1(T))*v1(T)*u3(T)*μ - (1//2)*Differential(T)(v1(T))*v1(T)*u2(T)*μ - (1//4)*(v2(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v2(T)*v1(T)*Differential(T)(u2(T))*μ - (1//2)*v2(T)*v1(T)*Differential(T)(u1(T))*μ + v2(T)*u3(T)*Differential(T)(v2(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(u2(T))*μ + (1//2)*v1(T)*u2(T)*Differential(T)(v2(T))*μ + Differential(T)(u2(T))*u3(T)*u2(T)*μ + (1//4)*Differential(T)(u1(T))*(u2(T)^2)*μ + (5//2)*(u1(T)^2)*v3(T)*μ*ω_lc + (5//4)*(u1(T)^2)*v2(T)*μ*ω_lc + (5//2)*u1(T)*v2(T)*u2(T)*μ*ω_lc + (5//2)*u1(T)*v1(T)*u2(T)*μ*ω_lc + (5//4)*(v3(T)^3)*μ*ω_lc + (5//2)*v3(T)*(v2(T)^2)*μ*ω_lc + (5//2)*v3(T)*(v1(T)^2)*μ*ω_lc + (5//4)*v3(T)*(u3(T)^2)*μ*ω_lc + (5//2)*v3(T)*(u2(T)^2)*μ*ω_lc + (5//4)*(v2(T)^2)*v1(T)*μ*ω_lc - (5//4)*v2(T)*(v1(T)^2)*μ*ω_lc - (5//4)*v1(T)*(u2(T)^2)*μ*ω_lc ~ 0
+
+v3(T) - (10//1)*Differential(T)(u3(T))*ω_lc - Differential(T)(v3(T))*μ - (25//1)*v3(T)*(ω_lc^2) + (5//1)*u3(T)*μ*ω_lc + (1//2)*(u1(T)^2)*Differential(T)(v3(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(v2(T))*μ + u1(T)*v3(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*u1(T)*v2(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*v2(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*u2(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u3(T))*v3(T)*u3(T)*μ + (3//4)*Differential(T)(v3(T))*(v3(T)^2)*μ + (1//2)*Differential(T)(v3(T))*(v2(T)^2)*μ + (1//2)*Differential(T)(v3(T))*(v1(T)^2)*μ + (1//4)*Differential(T)(v3(T))*(u3(T)^2)*μ + (1//2)*Differential(T)(v3(T))*(u2(T)^2)*μ + v3(T)*Differential(T)(v1(T))*v1(T)*μ + v3(T)*v2(T)*Differential(T)(v2(T))*μ + v3(T)*Differential(T)(u2(T))*u2(T)*μ + (1//4)*Differential(T)(v1(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(v1(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v1(T))*(u2(T)^2)*μ + (1//2)*v2(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*v2(T)*Differential(T)(u1(T))*u2(T)*μ - (1//4)*(v1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*v1(T)*Differential(T)(u2(T))*u2(T)*μ + (1//2)*v1(T)*Differential(T)(u1(T))*u2(T)*μ - (5//2)*(u1(T)^2)*u3(T)*μ*ω_lc - (5//4)*(u1(T)^2)*u2(T)*μ*ω_lc + (5//4)*u1(T)*(v2(T)^2)*μ*ω_lc + (5//2)*u1(T)*v2(T)*v1(T)*μ*ω_lc - (5//4)*u1(T)*(u2(T)^2)*μ*ω_lc - (5//4)*(v3(T)^2)*u3(T)*μ*ω_lc - (5//2)*(v2(T)^2)*u3(T)*μ*ω_lc - (5//2)*v2(T)*v1(T)*u2(T)*μ*ω_lc - (5//2)*(v1(T)^2)*u3(T)*μ*ω_lc + (5//4)*(v1(T)^2)*u2(T)*μ*ω_lc - (5//4)*(u3(T)^3)*μ*ω_lc - (5//2)*u3(T)*(u2(T)^2)*μ*ω_lc ~ 0
    `,4)),t("p",null,[s[15]||(s[15]=a("So far, ")),t("mjx-container",v,[(l(),e("svg",x,s[13]||(s[13]=[i('',1)]))),s[14]||(s[14]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"l"),t("mi",null,"c")])])])],-1))]),s[16]||(s[16]=a(" appears as any other harmonic. However, it is not fixed by any external drive or 'clock', instead, it emerges out of a Hopf instability in the system. We can verify that fixing ")),s[17]||(s[17]=t("code",null,"ω_lc",-1)),s[18]||(s[18]=a(" and calling ")),s[19]||(s[19]=t("code",null,"get_steady_states",-1)),s[20]||(s[20]=a("."))]),s[105]||(s[105]=i('
    julia
    get_steady_states(harmonic_eq, μ => 1:0.1:5, ω_lc => 1.2)

    gives a single solution with zero amplitude.

    ',2)),t("p",null,[s[23]||(s[23]=a("Taking instead ")),t("mjx-container",w,[(l(),e("svg",E,s[21]||(s[21]=[i('',1)]))),s[22]||(s[22]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"l"),t("mi",null,"c")])])])],-1))]),s[24]||(s[24]=a(" as a variable to be solved for ")),s[25]||(s[25]=t("a",{href:"/HarmonicBalance.jl/v0.10.7/background/limit_cycles#limit_cycles_bg"},"results in a phase freedom",-1)),s[26]||(s[26]=a(", implying an infinite number of solutions. To perform the ")),s[27]||(s[27]=t("a",{href:"/HarmonicBalance.jl/v0.10.7/background/limit_cycles#gauge_fixing"},"gauge-fixing procedure",-1)),s[28]||(s[28]=a(", we call ")),s[29]||(s[29]=t("code",null,"get_limit_cycles",-1)),s[30]||(s[30]=a(", marking the limit cycle harmonic as a keyword argument,"))]),s[106]||(s[106]=i(`
    julia
    result = get_limit_cycles(harmonic_eq, μ => 1:0.1:5, (), ω_lc)
    A steady state result for 41 parameter points
+
+Solution branches:   100
+   of which real:    4
+   of which stable:  4
+
+Classes: unique_cycle, stable, physical, Hopf, binary_labels

    The results show a fourfold degeneracy of solutions:

    julia
    plot(result, y="ω_lc")

    The automatically created solution class unique_cycle filters the degeneracy out:

    julia
    plot(result, y="ω_lc", class="unique_cycle")

    Driven system - coupled Duffings ​

    So far, we have largely focused on finding and analysing steady states, i.e., fixed points of the harmonic equations, which satisfy

    ',10)),t("mjx-container",H,[(l(),e("svg",D,s[31]||(s[31]=[i('',1)]))),s[32]||(s[32]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mfrac",null,[t("mrow",null,[t("mi",null,"d"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"u")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")]),t("mrow",null,[t("mi",null,"d"),t("mi",null,"T")])]),t("mo",null,"="),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"F")]),t("mo",{stretchy:"false"},"¯")])]),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"u")]),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mn",null,"0"),t("mstyle",{scriptlevel:"0"},[t("mspace",{width:"0.167em"})]),t("mo",null,".")])],-1))]),t("p",null,[s[37]||(s[37]=a("Fixed points are however merely a subset of possible solutions of Eq. \\eqref{eq:harmeqfull} – strictly speaking, solutions where ")),t("mjx-container",b,[(l(),e("svg",F,s[33]||(s[33]=[i('',1)]))),s[34]||(s[34]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"u")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),s[38]||(s[38]=a(" remains time-dependent are allowed. These are quite unusual, since ")),t("mjx-container",M,[(l(),e("svg",L,s[35]||(s[35]=[i('',1)]))),s[36]||(s[36]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"F")]),t("mo",{stretchy:"false"},"¯")])]),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"u")]),t("mo",{stretchy:"false"},")")])],-1))]),s[39]||(s[39]=a()),s[40]||(s[40]=t("a",{href:"/HarmonicBalance.jl/v0.10.7/background/harmonic_balance#intro_hb"},"is by construction time-independent",-1)),s[41]||(s[41]=a(" and Eq. \\eqref{eq:harmeqfull} thus possesses ")),s[42]||(s[42]=t("em",null,"continuous time-translation symmetry",-1)),s[43]||(s[43]=a(". The appearance of explicitly time-dependent solutions then consitutes spontaneous time-translation symmetry breaking."))]),t("p",null,[s[46]||(s[46]=a("Such solutions, known as ")),s[47]||(s[47]=t("em",null,"limit cycles",-1)),s[48]||(s[48]=a(", typically appear as closed periodic trajectories of the harmonic variables ")),t("mjx-container",V,[(l(),e("svg",C,s[44]||(s[44]=[i('',1)]))),s[45]||(s[45]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"u")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),s[49]||(s[49]=a(". The simplest way to numerically characterise them is a time-dependent simulation, using a steady-state diagram as a guide."))]),t("p",null,[s[54]||(s[54]=a("Here we reconstruct the results of ")),s[55]||(s[55]=t("a",{href:"https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.023526",target:"_blank",rel:"noreferrer"},"Zambon et al., Phys Rev. A 102, 023526 (2020)",-1)),s[56]||(s[56]=a(", where limit cycles are shown to appear in a system of two coupled nonlinear oscillators. In this problem, two oscillators ")),t("mjx-container",Z,[(l(),e("svg",j,s[50]||(s[50]=[i('',1)]))),s[51]||(s[51]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mn",null,"1")])])],-1))]),s[57]||(s[57]=a(" and ")),t("mjx-container",B,[(l(),e("svg",A,s[52]||(s[52]=[i('',1)]))),s[53]||(s[53]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mn",null,"2")])])],-1))]),s[58]||(s[58]=a(", have (the same) damping and Kerr nonlinearity and are linearly coupled,"))]),t("mjx-container",_,[(l(),e("svg",q,s[59]||(s[59]=[i('',1)]))),s[60]||(s[60]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[t("mtr",null,[t("mtd",null,[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mn",null,"1")]),t("mo",null,"+"),t("mi",null,"γ"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mn",null,"1")]),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"ω"),t("mn",null,"0"),t("mn",null,"2")]),t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",null,"+"),t("mi",null,"α"),t("msubsup",null,[t("mi",null,"x"),t("mn",null,"1"),t("mn",null,"3")]),t("mo",null,"+"),t("mn",null,"2"),t("mi",null,"J"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",null,"−"),t("msub",null,[t("mi",null,"x"),t("mn",null,"2")]),t("mo",{stretchy:"false"},")")]),t("mtd",null,[t("mi"),t("mo",null,"="),t("msub",null,[t("mi",null,"F"),t("mn",null,"0")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])]),t("mtr",null,[t("mtd",null,[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mn",null,"2")]),t("mo",null,"+"),t("mi",null,"γ"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mn",null,"2")]),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"ω"),t("mn",null,"0"),t("mn",null,"2")]),t("msub",null,[t("mi",null,"x"),t("mn",null,"2")]),t("mo",null,"+"),t("mi",null,"α"),t("msubsup",null,[t("mi",null,"x"),t("mn",null,"2"),t("mn",null,"3")]),t("mo",null,"+"),t("mn",null,"2"),t("mi",null,"J"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"x"),t("mn",null,"2")]),t("mo",null,"−"),t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",{stretchy:"false"},")")]),t("mtd",null,[t("mi"),t("mo",null,"="),t("mi",null,"η"),t("msub",null,[t("mi",null,"F"),t("mn",null,"0")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])])])])],-1))]),s[107]||(s[107]=i(`
    julia
    using HarmonicBalance
+@variables γ F α ω0 F0 η ω J t x(t) y(t);
+eqs = [d(x,t,2) + γ*d(x,t) + ω0^2*x + α*x^3+ 2*J*ω0*(x-y) - F0*cos(ω*t),
+       d(y,t,2) + γ * d(y,t) + ω0^2 * y + α*y^3 + 2*J*ω0*(y-x) - η*F0*cos(ω*t)]
+diff_eq = DifferentialEquation(eqs, [x,y])
    System of 2 differential equations
+Variables:       x(t), y(t)
+Harmonic ansatz: x(t) => ;   y(t) => ;   
+
+Differential(t)(Differential(t)(x(t))) - F0*cos(t*ω) + Differential(t)(x(t))*γ + 2J*(x(t) - y(t))*ω0 + x(t)*(ω0^2) + (x(t)^3)*α ~ 0
+Differential(t)(Differential(t)(y(t))) + Differential(t)(y(t))*γ - F0*cos(t*ω)*η + 2J*(-x(t) + y(t))*ω0 + y(t)*(ω0^2) + (y(t)^3)*α ~ 0
    `,2)),t("p",null,[s[65]||(s[65]=a("The analysis of Zambon et al. uses a frame rotating at the pump frequency ")),t("mjx-container",J,[(l(),e("svg",O,s[61]||(s[61]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[62]||(s[62]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),s[66]||(s[66]=a(" to describe both oscillators. For us, this means we expand both modes using ")),t("mjx-container",S,[(l(),e("svg",R,s[63]||(s[63]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[64]||(s[64]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),s[67]||(s[67]=a(" to obtain the harmonic equations."))]),s[108]||(s[108]=i(`
    julia
    add_harmonic!(diff_eq, x, ω)
+add_harmonic!(diff_eq, y, ω)
+
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 4 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T)
+Parameters: ω, ω0, J, α, γ, F0, η
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+y(t) = u2(T)*cos(ωt) + v2(T)*sin(ωt)
+
+Harmonic equations:
+
+-F0 + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ + (2//1)*J*u1(T)*ω0 - (2//1)*J*u2(T)*ω0 - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ 0
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - (2//1)*J*v2(T)*ω0 + (2//1)*J*v1(T)*ω0 - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0
+
+-F0*η + Differential(T)(u2(T))*γ + (2//1)*Differential(T)(v2(T))*ω - (2//1)*J*u1(T)*ω0 + (2//1)*J*u2(T)*ω0 + v2(T)*γ*ω - u2(T)*(ω^2) + u2(T)*(ω0^2) + (3//4)*(v2(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0
+
+-(2//1)*Differential(T)(u2(T))*ω + Differential(T)(v2(T))*γ + (2//1)*J*v2(T)*ω0 - (2//1)*J*v1(T)*ω0 - v2(T)*(ω^2) + v2(T)*(ω0^2) - u2(T)*γ*ω + (3//4)*(v2(T)^3)*α + (3//4)*v2(T)*(u2(T)^2)*α ~ 0
    `,2)),t("p",null,[s[70]||(s[70]=a("Solving for a range of drive amplitudes ")),t("mjx-container",z,[(l(),e("svg",G,s[68]||(s[68]=[i('',1)]))),s[69]||(s[69]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")])])],-1))]),s[71]||(s[71]=a(","))]),s[109]||(s[109]=i(`
    julia
    fixed = (
+    ω0 => 1.4504859, # natural frequency of separate modes (in paper's notation, ħω0 - J)
+    γ => 27.4e-6,    # damping
+    J => 154.1e-6,   # coupling term
+    α => 3.867e-7,   # Kerr nonlinearity
+    ω => 1.4507941,  # pump frequency, resonant with antisymmetric mode (in paper, ħω0 + J)
+    η => -0.08,      # pumping leaking to site 2  (F2 = ηF1)
+    F0 => 0.002       # pump amplitude (overriden in sweeps)
+)
+varied = F0 => range(0.002, 0.03, 50)
+
+result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 50 parameter points
+
+Solution branches:   9
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels

    Let us first see the steady states.

    julia
    p1 = plot(result, "u1^2 + v1^2", legend=false)
+p2 = plot(result, "u2^2 + v2^2")
+plot(p1, p2)

    ',5)),t("p",null,[s[78]||(s[78]=a("According to Zambon et al., a limit cycle solution exists around ")),t("mjx-container",P,[(l(),e("svg",X,s[72]||(s[72]=[i('',1)]))),s[73]||(s[73]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")]),t("mo",null,"≅"),t("mn",null,"0.011")])],-1))]),s[79]||(s[79]=a(", which can be accessed by a jump from branch 1 in an upwards sweep of ")),t("mjx-container",N,[(l(),e("svg",W,s[74]||(s[74]=[i('',1)]))),s[75]||(s[75]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")])])],-1))]),s[80]||(s[80]=a(". Since a limit cycle is not a steady state of our harmonic equations, it does not appear in the diagram. We do however see that branch 1 ceases to be stable around ")),t("mjx-container",I,[(l(),e("svg",$,s[76]||(s[76]=[i('',1)]))),s[77]||(s[77]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")]),t("mo",null,"≅"),t("mn",null,"0.010")])],-1))]),s[81]||(s[81]=a(", meaning a jump should occur."))]),t("p",null,[s[86]||(s[86]=a("Let us try and simulate the limit cycle. We could in principle run a time-dependent simulation with a fixed value of ")),t("mjx-container",K,[(l(),e("svg",U,s[82]||(s[82]=[i('',1)]))),s[83]||(s[83]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")])])],-1))]),s[87]||(s[87]=a(", but this would require a suitable initial condition. Instead, we will sweep ")),t("mjx-container",Y,[(l(),e("svg",t1,s[84]||(s[84]=[i('',1)]))),s[85]||(s[85]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")])])],-1))]),s[88]||(s[88]=a(" upwards from a low starting value. To observe the dynamics just after the jump has occurred, we follow the sweep by a time interval where the system evolves under fixed parameters."))]),s[110]||(s[110]=i(`
    julia
    using OrdinaryDiffEq
+initial_state = result[1][1]
+
+T = 2e6
+sweep = ParameterSweep(F0 => (0.002, 0.011), (0,T))
+
+# start from initial_state, use sweep, total time is 2*T
+time_problem = ODEProblem(harmonic_eq, initial_state, sweep=sweep, timespan=(0,2*T))
+time_evo = solve(time_problem, saveat=100);

    Inspecting the amplitude as a function of time,

    julia
    plot(time_evo, "sqrt(u1^2 + v1^2)", harmonic_eq)

    ',4)),t("p",null,[s[93]||(s[93]=a("we see that initially the sweep is adiabatic as it proceeds along the steady-state branch 1. At around ")),t("mjx-container",s1,[(l(),e("svg",a1,s[89]||(s[89]=[i('',1)]))),s[90]||(s[90]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"T"),t("mo",null,"="),t("mn",null,"2"),t("mi",null,"e"),t("mn",null,"6")])],-1))]),s[94]||(s[94]=a(", an instability occurs and ")),t("mjx-container",i1,[(l(),e("svg",e1,s[91]||(s[91]=[i('',1)]))),s[92]||(s[92]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),s[95]||(s[95]=a(" starts to rapidly oscillate. At that point, the sweep is stopped. Under free time evolution, the system then settles into a limit-cycle solution where the coordinates move along closed trajectories."))]),t("p",null,[s[100]||(s[100]=a("By plotting the ")),t("mjx-container",l1,[(l(),e("svg",n1,s[96]||(s[96]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D462",d:"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),s[97]||(s[97]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u")])],-1))]),s[101]||(s[101]=a(" and ")),t("mjx-container",T1,[(l(),e("svg",Q1,s[98]||(s[98]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D463",d:"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z",style:{"stroke-width":"3"}})])])],-1)]))),s[99]||(s[99]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"v")])],-1))]),s[102]||(s[102]=a(" variables against each other, we observe the limit cycle shapes in phase space,"))]),s[111]||(s[111]=i(`
    julia
    p1 = plot(time_evo, ["u1", "v1"], harmonic_eq)
+p2 = plot(time_evo, ["u2", "v2"], harmonic_eq)
+plot(p1, p2)

    ',2))])}const g1=n(d,[["render",r1]]);export{u1 as __pageData,g1 as default}; diff --git a/v0.10.7/assets/tutorials_limit_cycles.md.8x6X1DL_.lean.js b/v0.10.7/assets/tutorials_limit_cycles.md.8x6X1DL_.lean.js new file mode 100644 index 00000000..82b17104 --- /dev/null +++ b/v0.10.7/assets/tutorials_limit_cycles.md.8x6X1DL_.lean.js @@ -0,0 +1,91 @@ +import{_ as n,c as e,a4 as i,j as t,a,o as l}from"./chunks/framework.D3a-l5sU.js";const T="/HarmonicBalance.jl/v0.10.7/assets/oupuyzd.WSAVTkDY.png",Q="/HarmonicBalance.jl/v0.10.7/assets/nskzule.B3J9_Und.png",r="/HarmonicBalance.jl/v0.10.7/assets/noooogl.BsfZD08c.png",o="/HarmonicBalance.jl/v0.10.7/assets/deafupc.CP9uumgf.png",p="/HarmonicBalance.jl/v0.10.7/assets/zlfivvs.BsjSHmjf.png",u1=JSON.parse('{"title":"Limit 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    Limit cycles ​

    In contrast to the previous tutorials, limit cycle problems feature harmonic(s) whose numerical value is not imposed externally. We shall construct our HarmonicEquation as usual, but identify this harmonic as an extra variable, rather than a fixed parameter.

    Non-driven system - the van der Pol oscillator ​

    Here we solve the equation of motion of the van der Pol oscillator. This is a single-variable second-order ODE with continuous time-translation symmetry (i.e., no 'clock' imposing a frequency and/or phase), which displays periodic solutions known as relaxation oscillations. For more detail, refer also to arXiv:2308.06092.

    julia
    using HarmonicBalance
+@variables ω_lc, t, ω0, x(t), μ
+diff_eq = DifferentialEquation(d(d(x,t),t) - μ*(1-x^2) * d(x,t) + x, x)
    System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ;   
+
+x(t) + Differential(t)(Differential(t)(x(t))) - (1 - (x(t)^2))*Differential(t)(x(t))*μ ~ 0
    `,6)),t("p",null,[s[8]||(s[8]=a("Choosing to expand the motion of ")),t("mjx-container",h,[(l(),e("svg",m,s[0]||(s[0]=[i('',1)]))),s[1]||(s[1]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),s[9]||(s[9]=a(" using ")),t("mjx-container",k,[(l(),e("svg",u,s[2]||(s[2]=[i('',1)]))),s[3]||(s[3]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"l"),t("mi",null,"c")])])])],-1))]),s[10]||(s[10]=a(", ")),t("mjx-container",g,[(l(),e("svg",c,s[4]||(s[4]=[i('',1)]))),s[5]||(s[5]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"l"),t("mi",null,"c")])])])],-1))]),s[11]||(s[11]=a(" and ")),t("mjx-container",f,[(l(),e("svg",y,s[6]||(s[6]=[i('',1)]))),s[7]||(s[7]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"5"),t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"l"),t("mi",null,"c")])])])],-1))]),s[12]||(s[12]=a(", we define"))]),s[104]||(s[104]=i(`
    julia
    foreach(1:2:5) do i
+  add_harmonic!(diff_eq, x, i*ω_lc)
+end;

    and obtain 6 harmonic equations,

    julia
    harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 6 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T), u3(T), v3(T)
+Parameters: ω_lc, μ
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ω_lct) + v1(T)*sin(ω_lct) + u2(T)*cos(3ω_lct) + v2(T)*sin(3ω_lct) + u3(T)*cos(5ω_lct) + v3(T)*sin(5ω_lct)
+
+Harmonic equations:
+
+u1(T) + (2//1)*Differential(T)(v1(T))*ω_lc - Differential(T)(u1(T))*μ - u1(T)*(ω_lc^2) - v1(T)*μ*ω_lc + (1//4)*(u1(T)^2)*Differential(T)(u2(T))*μ + (3//4)*(u1(T)^2)*Differential(T)(u1(T))*μ + u1(T)*Differential(T)(u3(T))*u3(T)*μ + (1//2)*u1(T)*Differential(T)(u3(T))*u2(T)*μ + u1(T)*Differential(T)(v3(T))*v3(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*v2(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*Differential(T)(v1(T))*v2(T)*μ + (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*μ + u1(T)*v2(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*Differential(T)(u2(T))*u3(T)*μ + u1(T)*Differential(T)(u2(T))*u2(T)*μ + (1//2)*u1(T)*Differential(T)(u1(T))*u2(T)*μ - (1//4)*Differential(T)(u3(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(u3(T))*v2(T)*v1(T)*μ + (1//4)*Differential(T)(u3(T))*(u2(T)^2)*μ + (1//2)*Differential(T)(v3(T))*v2(T)*u2(T)*μ + (1//2)*Differential(T)(v3(T))*v1(T)*u2(T)*μ + (1//2)*(v3(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v3(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*v3(T)*v2(T)*Differential(T)(u2(T))*μ + (1//2)*v3(T)*v2(T)*Differential(T)(u1(T))*μ + (1//2)*v3(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*v3(T)*u2(T)*Differential(T)(v2(T))*μ - (1//2)*Differential(T)(v1(T))*v2(T)*u3(T)*μ - (1//2)*Differential(T)(v1(T))*v1(T)*u2(T)*μ + (1//2)*(v2(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v2(T)*v1(T)*Differential(T)(u1(T))*μ - (1//2)*v2(T)*u3(T)*Differential(T)(v2(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(u2(T))*μ + (1//4)*(v1(T)^2)*Differential(T)(u1(T))*μ - (1//2)*v1(T)*u3(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u2(T))*u3(T)*u2(T)*μ + (1//2)*(u3(T)^2)*Differential(T)(u1(T))*μ + (1//2)*u3(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*Differential(T)(u1(T))*(u2(T)^2)*μ + (1//4)*(u1(T)^2)*v2(T)*μ*ω_lc + (1//4)*(u1(T)^2)*v1(T)*μ*ω_lc + (1//2)*u1(T)*v3(T)*u2(T)*μ*ω_lc - (1//2)*u1(T)*v2(T)*u3(T)*μ*ω_lc - (1//2)*u1(T)*v1(T)*u2(T)*μ*ω_lc + (1//2)*(v3(T)^2)*v1(T)*μ*ω_lc + (1//4)*v3(T)*(v2(T)^2)*μ*ω_lc - (1//2)*v3(T)*v2(T)*v1(T)*μ*ω_lc - (1//4)*v3(T)*(u2(T)^2)*μ*ω_lc + (1//2)*(v2(T)^2)*v1(T)*μ*ω_lc - (1//4)*v2(T)*(v1(T)^2)*μ*ω_lc + (1//2)*v2(T)*u3(T)*u2(T)*μ*ω_lc + (1//4)*(v1(T)^3)*μ*ω_lc + (1//2)*v1(T)*(u3(T)^2)*μ*ω_lc - (1//2)*v1(T)*u3(T)*u2(T)*μ*ω_lc + (1//2)*v1(T)*(u2(T)^2)*μ*ω_lc ~ 0
+
+v1(T) - Differential(T)(v1(T))*μ - (2//1)*Differential(T)(u1(T))*ω_lc + u1(T)*μ*ω_lc - v1(T)*(ω_lc^2) + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*Differential(T)(u3(T))*v2(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*u2(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(u2(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*u1(T)*v2(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*u3(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u3(T))*v2(T)*u2(T)*μ + Differential(T)(u3(T))*v1(T)*u3(T)*μ - (1//2)*Differential(T)(u3(T))*v1(T)*u2(T)*μ + Differential(T)(v3(T))*v3(T)*v1(T)*μ + (1//4)*Differential(T)(v3(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(v3(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v3(T))*(u2(T)^2)*μ + (1//2)*(v3(T)^2)*Differential(T)(v1(T))*μ - (1//2)*v3(T)*Differential(T)(v1(T))*v2(T)*μ + (1//2)*v3(T)*v2(T)*Differential(T)(v2(T))*μ - (1//2)*v3(T)*v1(T)*Differential(T)(v2(T))*μ - (1//2)*v3(T)*Differential(T)(u2(T))*u2(T)*μ + (1//2)*v3(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*Differential(T)(v1(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(v1(T))*v2(T)*v1(T)*μ + (3//4)*Differential(T)(v1(T))*(v1(T)^2)*μ + (1//2)*Differential(T)(v1(T))*(u3(T)^2)*μ - (1//2)*Differential(T)(v1(T))*u3(T)*u2(T)*μ + (1//2)*Differential(T)(v1(T))*(u2(T)^2)*μ + v2(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*v2(T)*Differential(T)(u2(T))*u3(T)*μ - (1//2)*v2(T)*u3(T)*Differential(T)(u1(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*v1(T)*Differential(T)(u2(T))*u3(T)*μ + v1(T)*Differential(T)(u2(T))*u2(T)*μ - (1//2)*v1(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*u3(T)*u2(T)*Differential(T)(v2(T))*μ - (1//4)*(u1(T)^3)*μ*ω_lc - (1//4)*(u1(T)^2)*u2(T)*μ*ω_lc - (1//2)*u1(T)*(v3(T)^2)*μ*ω_lc - (1//2)*u1(T)*v3(T)*v2(T)*μ*ω_lc - (1//2)*u1(T)*(v2(T)^2)*μ*ω_lc - (1//2)*u1(T)*v2(T)*v1(T)*μ*ω_lc - (1//4)*u1(T)*(v1(T)^2)*μ*ω_lc - (1//2)*u1(T)*(u3(T)^2)*μ*ω_lc - (1//2)*u1(T)*u3(T)*u2(T)*μ*ω_lc - (1//2)*u1(T)*(u2(T)^2)*μ*ω_lc - (1//2)*v3(T)*v2(T)*u2(T)*μ*ω_lc - (1//2)*v3(T)*v1(T)*u2(T)*μ*ω_lc + (1//4)*(v2(T)^2)*u3(T)*μ*ω_lc + (1//2)*v2(T)*v1(T)*u3(T)*μ*ω_lc + (1//4)*(v1(T)^2)*u2(T)*μ*ω_lc - (1//4)*u3(T)*(u2(T)^2)*μ*ω_lc ~ 0
+
+u2(T) - Differential(T)(u2(T))*μ + (6//1)*Differential(T)(v2(T))*ω_lc - (3//1)*v2(T)*μ*ω_lc - (9//1)*u2(T)*(ω_lc^2) + (1//4)*(u1(T)^2)*Differential(T)(u3(T))*μ + (1//2)*(u1(T)^2)*Differential(T)(u2(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*Differential(T)(u3(T))*u2(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*v2(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*v1(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(v1(T))*μ + (1//2)*u1(T)*v3(T)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*μ + (1//2)*u1(T)*Differential(T)(u2(T))*u3(T)*μ + (1//2)*u1(T)*u3(T)*Differential(T)(u1(T))*μ + u1(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*Differential(T)(u3(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(u3(T))*(v1(T)^2)*μ + Differential(T)(u3(T))*u3(T)*u2(T)*μ + Differential(T)(v3(T))*v3(T)*u2(T)*μ - (1//2)*Differential(T)(v3(T))*v1(T)*u2(T)*μ + (1//2)*(v3(T)^2)*Differential(T)(u2(T))*μ - (1//2)*v3(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*v3(T)*v2(T)*Differential(T)(u1(T))*μ - (1//2)*v3(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*v3(T)*v1(T)*Differential(T)(u1(T))*μ + (1//2)*Differential(T)(v1(T))*v2(T)*u3(T)*μ - (1//2)*Differential(T)(v1(T))*v1(T)*u3(T)*μ + Differential(T)(v1(T))*v1(T)*u2(T)*μ + (1//4)*(v2(T)^2)*Differential(T)(u2(T))*μ + (1//2)*v2(T)*u2(T)*Differential(T)(v2(T))*μ + (1//2)*(v1(T)^2)*Differential(T)(u2(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v1(T)*u3(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u2(T))*(u3(T)^2)*μ + (3//4)*Differential(T)(u2(T))*(u2(T)^2)*μ + (1//2)*u3(T)*Differential(T)(u1(T))*u2(T)*μ + (3//4)*(u1(T)^2)*v3(T)*μ*ω_lc + (3//2)*(u1(T)^2)*v2(T)*μ*ω_lc + (3//4)*(u1(T)^2)*v1(T)*μ*ω_lc + (3//2)*u1(T)*v3(T)*u2(T)*μ*ω_lc - (3//2)*u1(T)*v2(T)*u3(T)*μ*ω_lc - (3//2)*u1(T)*v1(T)*u3(T)*μ*ω_lc + (3//2)*(v3(T)^2)*v2(T)*μ*ω_lc + (3//2)*v3(T)*v2(T)*v1(T)*μ*ω_lc - (3//4)*v3(T)*(v1(T)^2)*μ*ω_lc + (3//4)*(v2(T)^3)*μ*ω_lc + (3//2)*v2(T)*(v1(T)^2)*μ*ω_lc + (3//2)*v2(T)*(u3(T)^2)*μ*ω_lc + (3//4)*v2(T)*(u2(T)^2)*μ*ω_lc - (1//4)*(v1(T)^3)*μ*ω_lc + (3//2)*v1(T)*u3(T)*u2(T)*μ*ω_lc ~ 0
+
+v2(T) - (6//1)*Differential(T)(u2(T))*ω_lc - Differential(T)(v2(T))*μ - (9//1)*v2(T)*(ω_lc^2) + (3//1)*u2(T)*μ*ω_lc + (1//4)*(u1(T)^2)*Differential(T)(v3(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*μ + (1//2)*(u1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*Differential(T)(u3(T))*v2(T)*μ - (1//2)*u1(T)*Differential(T)(u3(T))*v1(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*u2(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*v3(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*u3(T)*μ + u1(T)*v2(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*u3(T)*Differential(T)(v2(T))*μ + Differential(T)(u3(T))*v2(T)*u3(T)*μ + (1//2)*Differential(T)(u3(T))*v1(T)*u2(T)*μ + Differential(T)(v3(T))*v3(T)*v2(T)*μ + (1//2)*Differential(T)(v3(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v3(T))*(v1(T)^2)*μ + (1//2)*(v3(T)^2)*Differential(T)(v2(T))*μ + (1//2)*v3(T)*Differential(T)(v1(T))*v2(T)*μ - (1//2)*v3(T)*Differential(T)(v1(T))*v1(T)*μ + (1//2)*v3(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*v3(T)*Differential(T)(u1(T))*u2(T)*μ + Differential(T)(v1(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v1(T))*(v1(T)^2)*μ + (1//2)*Differential(T)(v1(T))*u3(T)*u2(T)*μ + (3//4)*(v2(T)^2)*Differential(T)(v2(T))*μ + (1//2)*v2(T)*Differential(T)(u2(T))*u2(T)*μ - (1//2)*v2(T)*u3(T)*Differential(T)(u1(T))*μ + (1//2)*(v1(T)^2)*Differential(T)(v2(T))*μ + (1//2)*v1(T)*Differential(T)(u2(T))*u3(T)*μ - (1//2)*v1(T)*u3(T)*Differential(T)(u1(T))*μ + (1//2)*(u3(T)^2)*Differential(T)(v2(T))*μ + (1//4)*(u2(T)^2)*Differential(T)(v2(T))*μ - (1//4)*(u1(T)^3)*μ*ω_lc - (3//4)*(u1(T)^2)*u3(T)*μ*ω_lc - (3//2)*(u1(T)^2)*u2(T)*μ*ω_lc - (3//2)*u1(T)*v3(T)*v2(T)*μ*ω_lc - (3//2)*u1(T)*v3(T)*v1(T)*μ*ω_lc + (3//4)*u1(T)*(v1(T)^2)*μ*ω_lc - (3//2)*u1(T)*u3(T)*u2(T)*μ*ω_lc - (3//2)*(v3(T)^2)*u2(T)*μ*ω_lc + (3//2)*v3(T)*v1(T)*u2(T)*μ*ω_lc - (3//4)*(v2(T)^2)*u2(T)*μ*ω_lc - (3//2)*v2(T)*v1(T)*u3(T)*μ*ω_lc + (3//4)*(v1(T)^2)*u3(T)*μ*ω_lc - (3//2)*(v1(T)^2)*u2(T)*μ*ω_lc - (3//2)*(u3(T)^2)*u2(T)*μ*ω_lc - (3//4)*(u2(T)^3)*μ*ω_lc ~ 0
+
+u3(T) - Differential(T)(u3(T))*μ + (10//1)*Differential(T)(v3(T))*ω_lc - (5//1)*v3(T)*μ*ω_lc - (25//1)*u3(T)*(ω_lc^2) + (1//2)*(u1(T)^2)*Differential(T)(u3(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(u2(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*v2(T)*μ - (1//2)*u1(T)*v2(T)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*Differential(T)(u2(T))*u2(T)*μ + u1(T)*u3(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*Differential(T)(u1(T))*u2(T)*μ + (1//4)*Differential(T)(u3(T))*(v3(T)^2)*μ + (1//2)*Differential(T)(u3(T))*(v2(T)^2)*μ + (1//2)*Differential(T)(u3(T))*(v1(T)^2)*μ + (3//4)*Differential(T)(u3(T))*(u3(T)^2)*μ + (1//2)*Differential(T)(u3(T))*(u2(T)^2)*μ + (1//2)*Differential(T)(v3(T))*v3(T)*u3(T)*μ + (1//2)*Differential(T)(v1(T))*v2(T)*u2(T)*μ + Differential(T)(v1(T))*v1(T)*u3(T)*μ - (1//2)*Differential(T)(v1(T))*v1(T)*u2(T)*μ - (1//4)*(v2(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v2(T)*v1(T)*Differential(T)(u2(T))*μ - (1//2)*v2(T)*v1(T)*Differential(T)(u1(T))*μ + v2(T)*u3(T)*Differential(T)(v2(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(u2(T))*μ + (1//2)*v1(T)*u2(T)*Differential(T)(v2(T))*μ + Differential(T)(u2(T))*u3(T)*u2(T)*μ + (1//4)*Differential(T)(u1(T))*(u2(T)^2)*μ + (5//2)*(u1(T)^2)*v3(T)*μ*ω_lc + (5//4)*(u1(T)^2)*v2(T)*μ*ω_lc + (5//2)*u1(T)*v2(T)*u2(T)*μ*ω_lc + (5//2)*u1(T)*v1(T)*u2(T)*μ*ω_lc + (5//4)*(v3(T)^3)*μ*ω_lc + (5//2)*v3(T)*(v2(T)^2)*μ*ω_lc + (5//2)*v3(T)*(v1(T)^2)*μ*ω_lc + (5//4)*v3(T)*(u3(T)^2)*μ*ω_lc + (5//2)*v3(T)*(u2(T)^2)*μ*ω_lc + (5//4)*(v2(T)^2)*v1(T)*μ*ω_lc - (5//4)*v2(T)*(v1(T)^2)*μ*ω_lc - (5//4)*v1(T)*(u2(T)^2)*μ*ω_lc ~ 0
+
+v3(T) - (10//1)*Differential(T)(u3(T))*ω_lc - Differential(T)(v3(T))*μ - (25//1)*v3(T)*(ω_lc^2) + (5//1)*u3(T)*μ*ω_lc + (1//2)*(u1(T)^2)*Differential(T)(v3(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(v2(T))*μ + u1(T)*v3(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*u1(T)*v2(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*v2(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*u2(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u3(T))*v3(T)*u3(T)*μ + (3//4)*Differential(T)(v3(T))*(v3(T)^2)*μ + (1//2)*Differential(T)(v3(T))*(v2(T)^2)*μ + (1//2)*Differential(T)(v3(T))*(v1(T)^2)*μ + (1//4)*Differential(T)(v3(T))*(u3(T)^2)*μ + (1//2)*Differential(T)(v3(T))*(u2(T)^2)*μ + v3(T)*Differential(T)(v1(T))*v1(T)*μ + v3(T)*v2(T)*Differential(T)(v2(T))*μ + v3(T)*Differential(T)(u2(T))*u2(T)*μ + (1//4)*Differential(T)(v1(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(v1(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v1(T))*(u2(T)^2)*μ + (1//2)*v2(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*v2(T)*Differential(T)(u1(T))*u2(T)*μ - (1//4)*(v1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*v1(T)*Differential(T)(u2(T))*u2(T)*μ + (1//2)*v1(T)*Differential(T)(u1(T))*u2(T)*μ - (5//2)*(u1(T)^2)*u3(T)*μ*ω_lc - (5//4)*(u1(T)^2)*u2(T)*μ*ω_lc + (5//4)*u1(T)*(v2(T)^2)*μ*ω_lc + (5//2)*u1(T)*v2(T)*v1(T)*μ*ω_lc - (5//4)*u1(T)*(u2(T)^2)*μ*ω_lc - (5//4)*(v3(T)^2)*u3(T)*μ*ω_lc - (5//2)*(v2(T)^2)*u3(T)*μ*ω_lc - (5//2)*v2(T)*v1(T)*u2(T)*μ*ω_lc - (5//2)*(v1(T)^2)*u3(T)*μ*ω_lc + (5//4)*(v1(T)^2)*u2(T)*μ*ω_lc - (5//4)*(u3(T)^3)*μ*ω_lc - (5//2)*u3(T)*(u2(T)^2)*μ*ω_lc ~ 0
    `,4)),t("p",null,[s[15]||(s[15]=a("So far, ")),t("mjx-container",v,[(l(),e("svg",x,s[13]||(s[13]=[i('',1)]))),s[14]||(s[14]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"l"),t("mi",null,"c")])])])],-1))]),s[16]||(s[16]=a(" appears as any other harmonic. However, it is not fixed by any external drive or 'clock', instead, it emerges out of a Hopf instability in the system. We can verify that fixing ")),s[17]||(s[17]=t("code",null,"ω_lc",-1)),s[18]||(s[18]=a(" and calling ")),s[19]||(s[19]=t("code",null,"get_steady_states",-1)),s[20]||(s[20]=a("."))]),s[105]||(s[105]=i('
    julia
    get_steady_states(harmonic_eq, μ => 1:0.1:5, ω_lc => 1.2)

    gives a single solution with zero amplitude.

    ',2)),t("p",null,[s[23]||(s[23]=a("Taking instead ")),t("mjx-container",w,[(l(),e("svg",E,s[21]||(s[21]=[i('',1)]))),s[22]||(s[22]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"l"),t("mi",null,"c")])])])],-1))]),s[24]||(s[24]=a(" as a variable to be solved for ")),s[25]||(s[25]=t("a",{href:"/HarmonicBalance.jl/v0.10.7/background/limit_cycles#limit_cycles_bg"},"results in a phase freedom",-1)),s[26]||(s[26]=a(", implying an infinite number of solutions. To perform the ")),s[27]||(s[27]=t("a",{href:"/HarmonicBalance.jl/v0.10.7/background/limit_cycles#gauge_fixing"},"gauge-fixing procedure",-1)),s[28]||(s[28]=a(", we call ")),s[29]||(s[29]=t("code",null,"get_limit_cycles",-1)),s[30]||(s[30]=a(", marking the limit cycle harmonic as a keyword argument,"))]),s[106]||(s[106]=i(`
    julia
    result = get_limit_cycles(harmonic_eq, μ => 1:0.1:5, (), ω_lc)
    A steady state result for 41 parameter points
+
+Solution branches:   100
+   of which real:    4
+   of which stable:  4
+
+Classes: unique_cycle, stable, physical, Hopf, binary_labels

    The results show a fourfold degeneracy of solutions:

    julia
    plot(result, y="ω_lc")

    The automatically created solution class unique_cycle filters the degeneracy out:

    julia
    plot(result, y="ω_lc", class="unique_cycle")

    Driven system - coupled Duffings ​

    So far, we have largely focused on finding and analysing steady states, i.e., fixed points of the harmonic equations, which satisfy

    ',10)),t("mjx-container",H,[(l(),e("svg",D,s[31]||(s[31]=[i('',1)]))),s[32]||(s[32]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mfrac",null,[t("mrow",null,[t("mi",null,"d"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"u")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")]),t("mrow",null,[t("mi",null,"d"),t("mi",null,"T")])]),t("mo",null,"="),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"F")]),t("mo",{stretchy:"false"},"¯")])]),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"u")]),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mn",null,"0"),t("mstyle",{scriptlevel:"0"},[t("mspace",{width:"0.167em"})]),t("mo",null,".")])],-1))]),t("p",null,[s[37]||(s[37]=a("Fixed points are however merely a subset of possible solutions of Eq. \\eqref{eq:harmeqfull} – strictly speaking, solutions where ")),t("mjx-container",b,[(l(),e("svg",F,s[33]||(s[33]=[i('',1)]))),s[34]||(s[34]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"u")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),s[38]||(s[38]=a(" remains time-dependent are allowed. These are quite unusual, since ")),t("mjx-container",M,[(l(),e("svg",L,s[35]||(s[35]=[i('',1)]))),s[36]||(s[36]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"F")]),t("mo",{stretchy:"false"},"¯")])]),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"u")]),t("mo",{stretchy:"false"},")")])],-1))]),s[39]||(s[39]=a()),s[40]||(s[40]=t("a",{href:"/HarmonicBalance.jl/v0.10.7/background/harmonic_balance#intro_hb"},"is by construction time-independent",-1)),s[41]||(s[41]=a(" and Eq. \\eqref{eq:harmeqfull} thus possesses ")),s[42]||(s[42]=t("em",null,"continuous time-translation symmetry",-1)),s[43]||(s[43]=a(". The appearance of explicitly time-dependent solutions then consitutes spontaneous time-translation symmetry breaking."))]),t("p",null,[s[46]||(s[46]=a("Such solutions, known as ")),s[47]||(s[47]=t("em",null,"limit cycles",-1)),s[48]||(s[48]=a(", typically appear as closed periodic trajectories of the harmonic variables ")),t("mjx-container",V,[(l(),e("svg",C,s[44]||(s[44]=[i('',1)]))),s[45]||(s[45]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"u")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),s[49]||(s[49]=a(". The simplest way to numerically characterise them is a time-dependent simulation, using a steady-state diagram as a guide."))]),t("p",null,[s[54]||(s[54]=a("Here we reconstruct the results of ")),s[55]||(s[55]=t("a",{href:"https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.023526",target:"_blank",rel:"noreferrer"},"Zambon et al., Phys Rev. A 102, 023526 (2020)",-1)),s[56]||(s[56]=a(", where limit cycles are shown to appear in a system of two coupled nonlinear oscillators. In this problem, two oscillators ")),t("mjx-container",Z,[(l(),e("svg",j,s[50]||(s[50]=[i('',1)]))),s[51]||(s[51]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mn",null,"1")])])],-1))]),s[57]||(s[57]=a(" and ")),t("mjx-container",B,[(l(),e("svg",A,s[52]||(s[52]=[i('',1)]))),s[53]||(s[53]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mn",null,"2")])])],-1))]),s[58]||(s[58]=a(", have (the same) damping and Kerr nonlinearity and are linearly coupled,"))]),t("mjx-container",_,[(l(),e("svg",q,s[59]||(s[59]=[i('',1)]))),s[60]||(s[60]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[t("mtr",null,[t("mtd",null,[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mn",null,"1")]),t("mo",null,"+"),t("mi",null,"γ"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mn",null,"1")]),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"ω"),t("mn",null,"0"),t("mn",null,"2")]),t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",null,"+"),t("mi",null,"α"),t("msubsup",null,[t("mi",null,"x"),t("mn",null,"1"),t("mn",null,"3")]),t("mo",null,"+"),t("mn",null,"2"),t("mi",null,"J"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",null,"−"),t("msub",null,[t("mi",null,"x"),t("mn",null,"2")]),t("mo",{stretchy:"false"},")")]),t("mtd",null,[t("mi"),t("mo",null,"="),t("msub",null,[t("mi",null,"F"),t("mn",null,"0")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])]),t("mtr",null,[t("mtd",null,[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mn",null,"2")]),t("mo",null,"+"),t("mi",null,"γ"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mn",null,"2")]),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"ω"),t("mn",null,"0"),t("mn",null,"2")]),t("msub",null,[t("mi",null,"x"),t("mn",null,"2")]),t("mo",null,"+"),t("mi",null,"α"),t("msubsup",null,[t("mi",null,"x"),t("mn",null,"2"),t("mn",null,"3")]),t("mo",null,"+"),t("mn",null,"2"),t("mi",null,"J"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"x"),t("mn",null,"2")]),t("mo",null,"−"),t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",{stretchy:"false"},")")]),t("mtd",null,[t("mi"),t("mo",null,"="),t("mi",null,"η"),t("msub",null,[t("mi",null,"F"),t("mn",null,"0")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])])])])],-1))]),s[107]||(s[107]=i(`
    julia
    using HarmonicBalance
+@variables γ F α ω0 F0 η ω J t x(t) y(t);
+eqs = [d(x,t,2) + γ*d(x,t) + ω0^2*x + α*x^3+ 2*J*ω0*(x-y) - F0*cos(ω*t),
+       d(y,t,2) + γ * d(y,t) + ω0^2 * y + α*y^3 + 2*J*ω0*(y-x) - η*F0*cos(ω*t)]
+diff_eq = DifferentialEquation(eqs, [x,y])
    System of 2 differential equations
+Variables:       x(t), y(t)
+Harmonic ansatz: x(t) => ;   y(t) => ;   
+
+Differential(t)(Differential(t)(x(t))) - F0*cos(t*ω) + Differential(t)(x(t))*γ + 2J*(x(t) - y(t))*ω0 + x(t)*(ω0^2) + (x(t)^3)*α ~ 0
+Differential(t)(Differential(t)(y(t))) + Differential(t)(y(t))*γ - F0*cos(t*ω)*η + 2J*(-x(t) + y(t))*ω0 + y(t)*(ω0^2) + (y(t)^3)*α ~ 0
    `,2)),t("p",null,[s[65]||(s[65]=a("The analysis of Zambon et al. uses a frame rotating at the pump frequency ")),t("mjx-container",J,[(l(),e("svg",O,s[61]||(s[61]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[62]||(s[62]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),s[66]||(s[66]=a(" to describe both oscillators. For us, this means we expand both modes using ")),t("mjx-container",S,[(l(),e("svg",R,s[63]||(s[63]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[64]||(s[64]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),s[67]||(s[67]=a(" to obtain the harmonic equations."))]),s[108]||(s[108]=i(`
    julia
    add_harmonic!(diff_eq, x, ω)
+add_harmonic!(diff_eq, y, ω)
+
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 4 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T)
+Parameters: ω, ω0, J, α, γ, F0, η
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+y(t) = u2(T)*cos(ωt) + v2(T)*sin(ωt)
+
+Harmonic equations:
+
+-F0 + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ + (2//1)*J*u1(T)*ω0 - (2//1)*J*u2(T)*ω0 - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ 0
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - (2//1)*J*v2(T)*ω0 + (2//1)*J*v1(T)*ω0 - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0
+
+-F0*η + Differential(T)(u2(T))*γ + (2//1)*Differential(T)(v2(T))*ω - (2//1)*J*u1(T)*ω0 + (2//1)*J*u2(T)*ω0 + v2(T)*γ*ω - u2(T)*(ω^2) + u2(T)*(ω0^2) + (3//4)*(v2(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0
+
+-(2//1)*Differential(T)(u2(T))*ω + Differential(T)(v2(T))*γ + (2//1)*J*v2(T)*ω0 - (2//1)*J*v1(T)*ω0 - v2(T)*(ω^2) + v2(T)*(ω0^2) - u2(T)*γ*ω + (3//4)*(v2(T)^3)*α + (3//4)*v2(T)*(u2(T)^2)*α ~ 0
    `,2)),t("p",null,[s[70]||(s[70]=a("Solving for a range of drive amplitudes ")),t("mjx-container",z,[(l(),e("svg",G,s[68]||(s[68]=[i('',1)]))),s[69]||(s[69]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")])])],-1))]),s[71]||(s[71]=a(","))]),s[109]||(s[109]=i(`
    julia
    fixed = (
+    ω0 => 1.4504859, # natural frequency of separate modes (in paper's notation, ħω0 - J)
+    γ => 27.4e-6,    # damping
+    J => 154.1e-6,   # coupling term
+    α => 3.867e-7,   # Kerr nonlinearity
+    ω => 1.4507941,  # pump frequency, resonant with antisymmetric mode (in paper, ħω0 + J)
+    η => -0.08,      # pumping leaking to site 2  (F2 = ηF1)
+    F0 => 0.002       # pump amplitude (overriden in sweeps)
+)
+varied = F0 => range(0.002, 0.03, 50)
+
+result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 50 parameter points
+
+Solution branches:   9
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels

    Let us first see the steady states.

    julia
    p1 = plot(result, "u1^2 + v1^2", legend=false)
+p2 = plot(result, "u2^2 + v2^2")
+plot(p1, p2)

    ',5)),t("p",null,[s[78]||(s[78]=a("According to Zambon et al., a limit cycle solution exists around ")),t("mjx-container",P,[(l(),e("svg",X,s[72]||(s[72]=[i('',1)]))),s[73]||(s[73]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")]),t("mo",null,"≅"),t("mn",null,"0.011")])],-1))]),s[79]||(s[79]=a(", which can be accessed by a jump from branch 1 in an upwards sweep of ")),t("mjx-container",N,[(l(),e("svg",W,s[74]||(s[74]=[i('',1)]))),s[75]||(s[75]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")])])],-1))]),s[80]||(s[80]=a(". Since a limit cycle is not a steady state of our harmonic equations, it does not appear in the diagram. We do however see that branch 1 ceases to be stable around ")),t("mjx-container",I,[(l(),e("svg",$,s[76]||(s[76]=[i('',1)]))),s[77]||(s[77]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")]),t("mo",null,"≅"),t("mn",null,"0.010")])],-1))]),s[81]||(s[81]=a(", meaning a jump should occur."))]),t("p",null,[s[86]||(s[86]=a("Let us try and simulate the limit cycle. We could in principle run a time-dependent simulation with a fixed value of ")),t("mjx-container",K,[(l(),e("svg",U,s[82]||(s[82]=[i('',1)]))),s[83]||(s[83]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")])])],-1))]),s[87]||(s[87]=a(", but this would require a suitable initial condition. Instead, we will sweep ")),t("mjx-container",Y,[(l(),e("svg",t1,s[84]||(s[84]=[i('',1)]))),s[85]||(s[85]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"F"),t("mn",null,"0")])])],-1))]),s[88]||(s[88]=a(" upwards from a low starting value. To observe the dynamics just after the jump has occurred, we follow the sweep by a time interval where the system evolves under fixed parameters."))]),s[110]||(s[110]=i(`
    julia
    using OrdinaryDiffEq
+initial_state = result[1][1]
+
+T = 2e6
+sweep = ParameterSweep(F0 => (0.002, 0.011), (0,T))
+
+# start from initial_state, use sweep, total time is 2*T
+time_problem = ODEProblem(harmonic_eq, initial_state, sweep=sweep, timespan=(0,2*T))
+time_evo = solve(time_problem, saveat=100);

    Inspecting the amplitude as a function of time,

    julia
    plot(time_evo, "sqrt(u1^2 + v1^2)", harmonic_eq)

    ',4)),t("p",null,[s[93]||(s[93]=a("we see that initially the sweep is adiabatic as it proceeds along the steady-state branch 1. At around ")),t("mjx-container",s1,[(l(),e("svg",a1,s[89]||(s[89]=[i('',1)]))),s[90]||(s[90]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"T"),t("mo",null,"="),t("mn",null,"2"),t("mi",null,"e"),t("mn",null,"6")])],-1))]),s[94]||(s[94]=a(", an instability occurs and ")),t("mjx-container",i1,[(l(),e("svg",e1,s[91]||(s[91]=[i('',1)]))),s[92]||(s[92]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"u"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")")])],-1))]),s[95]||(s[95]=a(" starts to rapidly oscillate. At that point, the sweep is stopped. Under free time evolution, the system then settles into a limit-cycle solution where the coordinates move along closed trajectories."))]),t("p",null,[s[100]||(s[100]=a("By plotting the ")),t("mjx-container",l1,[(l(),e("svg",n1,s[96]||(s[96]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D462",d:"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),s[97]||(s[97]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"u")])],-1))]),s[101]||(s[101]=a(" and ")),t("mjx-container",T1,[(l(),e("svg",Q1,s[98]||(s[98]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D463",d:"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z",style:{"stroke-width":"3"}})])])],-1)]))),s[99]||(s[99]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"v")])],-1))]),s[102]||(s[102]=a(" variables against each other, we observe the limit cycle shapes in phase space,"))]),s[111]||(s[111]=i(`
    julia
    p1 = plot(time_evo, ["u1", "v1"], harmonic_eq)
+p2 = plot(time_evo, ["u2", "v2"], harmonic_eq)
+plot(p1, p2)

    ',2))])}const g1=n(d,[["render",r1]]);export{u1 as __pageData,g1 as default}; diff --git a/v0.10.7/assets/tutorials_linear_response.md.ywigf0O0.js b/v0.10.7/assets/tutorials_linear_response.md.ywigf0O0.js new file mode 100644 index 00000000..224d9405 --- /dev/null +++ b/v0.10.7/assets/tutorials_linear_response.md.ywigf0O0.js @@ -0,0 +1,51 @@ +import{_ as l,c as t,a4 as n,j as i,a,o as e}from"./chunks/framework.D3a-l5sU.js";const h="/HarmonicBalance.jl/v0.10.7/assets/hgawaky.C1mRfhhg.png",p="/HarmonicBalance.jl/v0.10.7/assets/xumgmjw.C91AM-T5.png",r="/HarmonicBalance.jl/v0.10.7/assets/ltfrlnq.TE4cNA4T.png",k="/HarmonicBalance.jl/v0.10.7/assets/wstquaq.CF_iK7k1.png",o="/HarmonicBalance.jl/v0.10.7/assets/dvmccvh.D2avJWJQ.png",d="/HarmonicBalance.jl/v0.10.7/assets/badztxx.BKS8fzbs.png",g="/HarmonicBalance.jl/v0.10.7/assets/hfzgltm.DaP9_FvO.png",O=JSON.parse('{"title":"Linear response","description":"","frontmatter":{},"headers":[],"relativePath":"tutorials/linear_response.md","filePath":"tutorials/linear_response.md"}'),Q={name:"tutorials/linear_response.md"},E={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.206ex",height:"2.158ex",role:"img",focusable:"false",viewBox:"0 -871.8 4069.2 953.8","aria-hidden":"true"},y={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},T={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.357ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.34ex",height:"1.927ex",role:"img",focusable:"false",viewBox:"0 -694 1034.4 851.8","aria-hidden":"true"},u={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.375ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.395ex",height:"1.377ex",role:"img",focusable:"false",viewBox:"0 -443 1058.6 608.6","aria-hidden":"true"},x={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},F={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},w={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},C={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},v={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},f={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},B={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"12.474ex",height:"2.139ex",role:"img",focusable:"false",viewBox:"0 -863.3 5513.7 945.3","aria-hidden":"true"},D={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},H={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.375ex"},xmlns:"http://www.w3.org/2000/svg",width:"6.819ex",height:"1.694ex",role:"img",focusable:"false",viewBox:"0 -583 3014.1 748.6","aria-hidden":"true"},A={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},M={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.375ex"},xmlns:"http://www.w3.org/2000/svg",width:"6.819ex",height:"1.694ex",role:"img",focusable:"false",viewBox:"0 -583 3014.1 748.6","aria-hidden":"true"},L={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},j={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"1.695ex",height:"1.538ex",role:"img",focusable:"false",viewBox:"0 -680 749 680","aria-hidden":"true"},V={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"1.695ex",height:"1.538ex",role:"img",focusable:"false",viewBox:"0 -680 749 680","aria-hidden":"true"},Z={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},z={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"1.695ex",height:"1.538ex",role:"img",focusable:"false",viewBox:"0 -680 749 680","aria-hidden":"true"};function S(_,s,J,P,G,N){return e(),t("div",null,[s[50]||(s[50]=n(`

    Linear response ​

    In HarmonicBalance.jl, the stability and linear response are treated using the LinearResponse module.

    Here we calculate the white noise response of a simple nonlinear system. A set of reference results may be found in Huber et al. in Phys. Rev. X 10, 021066 (2020). We start by defining the Duffing oscillator

    julia
    using HarmonicBalance, Plots
+using Plots.Measures: mm
+@variables α, ω, ω0, F, γ, t, x(t); # declare constant variables and a function x(t)
+
+# define ODE
+diff_eq = DifferentialEquation(d(x,t,2) + ω0*x + α*x^3 + γ*d(x,t) ~ F*cos(ω*t), x)
+
+# specify the ansatz x = u(T) cos(ω*t) + v(T) sin(ω*t)
+add_harmonic!(diff_eq, x, ω)
+
+# implement ansatz to get harmonic equations
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω0, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+u1(T)*ω0 + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ F
+
+Differential(T)(v1(T))*γ + v1(T)*ω0 - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0

    Linear regime ​

    `,6)),i("p",null,[s[2]||(s[2]=a("When driven weakly, the Duffing resonator behaves quasi-linearly, i.e, its response to noise is independent of the applied drive. We see that for weak driving, ")),i("mjx-container",E,[(e(),t("svg",m,s[0]||(s[0]=[n('',1)]))),s[1]||(s[1]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"F"),i("mo",null,"="),i("msup",null,[i("mn",null,"10"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mo",null,"−"),i("mn",null,"4")])])])],-1))]),s[3]||(s[3]=a(", the amplitude is a Lorentzian."))]),s[51]||(s[51]=n(`
    julia
    fixed = (α => 1, ω0 => 1.0, γ => 0.005, F => 0.0001)   # fixed parameters
+varied = ω => range(0.95, 1.05, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
+
+plot(result, "sqrt(u1^2 + v1^2)")

    ',2)),i("p",null,[s[6]||(s[6]=a("To find the fluctuation on the top of the steady state one often employs a ")),s[7]||(s[7]=i("a",{href:"https://en.wikipedia.org/wiki/Linear_dynamical_system",target:"_blank",rel:"noreferrer"},"Bogoliubov-de Gennes analyses",-1)),s[8]||(s[8]=a(". Here, we compute the eigenvalues ")),i("mjx-container",y,[(e(),t("svg",T,s[4]||(s[4]=[n('',1)]))),s[5]||(s[5]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"λ"),i("mi",null,"k")])])],-1))]),s[9]||(s[9]=a(' of the Jacobian matrix at the steady state. The imaginary part of the eigenvalues gives characteristic frequencies of the "quasi-particle excitations". The real part gives the lifetime of these excitations.'))]),s[52]||(s[52]=n(`

    One can plot the eigenvalues as follows

    julia
    plot(
+    plot_eigenvalues(result, branch=1),
+    plot_eigenvalues(result, branch=1, type=:real, ylims=(-0.003, 0)),
+)

    We find a single pair of complex conjugate eigenvalues linearly changing with the driving frequency. Both real parts are negative, indicating stability.

    As discussed in background section on linear response, the excitation manifest itself as a lorentenzian peak in a power spectral density (PSD) measurement. The PSD can be plotted using plot_linear_response:

    julia
    plot_linear_response(result, x, Ω_range=range(0.95, 1.05, 300), branch=1, logscale=true)

    ',7)),i("p",null,[s[16]||(s[16]=a("The response has a peak at ")),i("mjx-container",u,[(e(),t("svg",c,s[10]||(s[10]=[n('',1)]))),s[11]||(s[11]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"ω"),i("mn",null,"0")])])],-1))]),s[17]||(s[17]=a(", irrespective of the driving frequency ")),i("mjx-container",x,[(e(),t("svg",F,s[12]||(s[12]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[13]||(s[13]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω")])],-1))]),s[18]||(s[18]=a(". Indeed, the eigenvalues shown before where plotted in the rotating frame at the frequency of the drive ")),i("mjx-container",w,[(e(),t("svg",C,s[14]||(s[14]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[15]||(s[15]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω")])],-1))]),s[19]||(s[19]=a(". Hence, the imaginary part of eigenvalues shows the frequency (energy) needed to excite the system at it natural frequency (The frequency its want to be excited at.)"))]),i("p",null,[s[22]||(s[22]=a('Note the slight "bending" of the noise peak with ')),i("mjx-container",v,[(e(),t("svg",f,s[20]||(s[20]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[21]||(s[21]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω")])],-1))]),s[23]||(s[23]=a(" - this is given by the failure of the first-order calculation to capture response far-detuned from the drive frequency."))]),s[53]||(s[53]=i("h3",{id:"Nonlinear-regime",tabindex:"-1"},[a("Nonlinear regime "),i("a",{class:"header-anchor",href:"#Nonlinear-regime","aria-label":'Permalink to "Nonlinear regime {#Nonlinear-regime}"'},"​")],-1)),i("p",null,[s[26]||(s[26]=a("For strong driving, matters get more complicated. Let us now use a drive ")),i("mjx-container",b,[(e(),t("svg",B,s[24]||(s[24]=[n('',1)]))),s[25]||(s[25]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"F"),i("mo",null,"="),i("mn",null,"2"),i("mo",null,"∗"),i("msup",null,[i("mn",null,"10"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mo",null,"−"),i("mn",null,"3")])])])],-1))]),s[27]||(s[27]=a(" :"))]),s[54]||(s[54]=n(`
    julia
    fixed = (α => 1, ω0 => 1.0, γ => 0.005, F => 0.002)   # fixed parameters
+varied = ω => range(0.95, 1.05, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
+
+plot(result, x="ω", y="sqrt(u1^2 + v1^2)");

    The amplitude is the well-known Duffing curve. Let's look at the eigenvalues of the two stable branches, 1 and 2.

    julia
    plot(
+    plot_eigenvalues(result, branch=1),
+    plot_eigenvalues(result, branch=1, type=:real, ylims=(-0.003, 0)),
+    plot_eigenvalues(result, branch=2),
+    plot_eigenvalues(result, branch=2, type=:real, ylims=(-0.003, 0)),
+)

    ',5)),i("p",null,[s[30]||(s[30]=a("Again every branch gives a single pair of complex conjugate eigenvalues. However, for branch 1, the characteristic frequencies due not change linearly with the driving frequency around ")),i("mjx-container",D,[(e(),t("svg",H,s[28]||(s[28]=[n('',1)]))),s[29]||(s[29]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω"),i("mo",null,"="),i("msub",null,[i("mi",null,"ω"),i("mn",null,"0")])])],-1))]),s[31]||(s[31]=a(". This is a sign of steady state becoming nonlinear at large amplitudes."))]),s[55]||(s[55]=n(`

    The same can be seen in the PSD:

    julia
    plot(
+  plot_linear_response(result, x, branch=1, Ω_range=range(0.95,1.1,300), logscale=true),
+  plot_linear_response(result, x, branch=2, Ω_range=range(0.9,1.1,300), logscale=true),
+    size=(600, 250), margin=3mm
+)

    In branch 1 the linear response to white noise shows more than one peak. This is a distinctly nonlinear phenomenon, indicitive if the squeezing of the steady state. Branch 2 is again quasi-linear, which stems from its low amplitude.

    ',4)),i("p",null,[s[36]||(s[36]=a("Following ")),s[37]||(s[37]=i("a",{href:"https://doi.org/10.1103/PhysRevX.10.021066",target:"_blank",rel:"noreferrer"},"Huber et al.",-1)),s[38]||(s[38]=a(", we may also fix ")),i("mjx-container",A,[(e(),t("svg",M,s[32]||(s[32]=[n('',1)]))),s[33]||(s[33]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω"),i("mo",null,"="),i("msub",null,[i("mi",null,"ω"),i("mn",null,"0")])])],-1))]),s[39]||(s[39]=a(" and plot the linear response as a function of ")),i("mjx-container",L,[(e(),t("svg",j,s[34]||(s[34]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D439",d:"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z",style:{"stroke-width":"3"}})])])],-1)]))),s[35]||(s[35]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"F")])],-1))]),s[40]||(s[40]=a(". The response turns out to be single-valued over a large range of driving strengths. Using a log scale for the x-axis:"))]),s[56]||(s[56]=n(`
    julia
    fixed = (α => 1., ω0 => 1.0, γ => 1e-2, ω => 1)   # fixed parameters
+swept = F => 10 .^ range(-6, -1, 200)           # range of parameter values
+result = get_steady_states(harmonic_eq, swept, fixed)
+
+plot(
+  plot(result, "sqrt(u1^2 + v1^2)", xscale=:log),
+  plot_linear_response(result, x, branch=1, Ω_range=range(0.9,1.1,300), logscale=true, xscale=:log),
+  size=(600, 250), margin=3mm
+)

    ',2)),i("p",null,[s[45]||(s[45]=a("We see that for low ")),i("mjx-container",V,[(e(),t("svg",q,s[41]||(s[41]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D439",d:"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z",style:{"stroke-width":"3"}})])])],-1)]))),s[42]||(s[42]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"F")])],-1))]),s[46]||(s[46]=a(", quasi-linear behaviour with a single Lorentzian response occurs, while for larger ")),i("mjx-container",Z,[(e(),t("svg",z,s[43]||(s[43]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D439",d:"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z",style:{"stroke-width":"3"}})])])],-1)]))),s[44]||(s[44]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"F")])],-1))]),s[47]||(s[47]=a(", two peaks form in the noise response. The two peaks are strongly unequal in magnitude, which is an example of internal squeezing (See supplemental material of ")),s[48]||(s[48]=i("a",{href:"https://doi.org/10.1103/PhysRevX.10.021066",target:"_blank",rel:"noreferrer"},"Huber et al.",-1)),s[49]||(s[49]=a(")."))])])}const W=l(Q,[["render",S]]);export{O as __pageData,W as default}; diff --git a/v0.10.7/assets/tutorials_linear_response.md.ywigf0O0.lean.js b/v0.10.7/assets/tutorials_linear_response.md.ywigf0O0.lean.js new file mode 100644 index 00000000..224d9405 --- /dev/null +++ b/v0.10.7/assets/tutorials_linear_response.md.ywigf0O0.lean.js @@ -0,0 +1,51 @@ +import{_ as l,c as t,a4 as n,j as i,a,o as e}from"./chunks/framework.D3a-l5sU.js";const 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    Linear response ​

    In HarmonicBalance.jl, the stability and linear response are treated using the LinearResponse module.

    Here we calculate the white noise response of a simple nonlinear system. A set of reference results may be found in Huber et al. in Phys. Rev. X 10, 021066 (2020). We start by defining the Duffing oscillator

    julia
    using HarmonicBalance, Plots
+using Plots.Measures: mm
+@variables α, ω, ω0, F, γ, t, x(t); # declare constant variables and a function x(t)
+
+# define ODE
+diff_eq = DifferentialEquation(d(x,t,2) + ω0*x + α*x^3 + γ*d(x,t) ~ F*cos(ω*t), x)
+
+# specify the ansatz x = u(T) cos(ω*t) + v(T) sin(ω*t)
+add_harmonic!(diff_eq, x, ω)
+
+# implement ansatz to get harmonic equations
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω0, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+u1(T)*ω0 + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ F
+
+Differential(T)(v1(T))*γ + v1(T)*ω0 - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0

    Linear regime ​

    `,6)),i("p",null,[s[2]||(s[2]=a("When driven weakly, the Duffing resonator behaves quasi-linearly, i.e, its response to noise is independent of the applied drive. We see that for weak driving, ")),i("mjx-container",E,[(e(),t("svg",m,s[0]||(s[0]=[n('',1)]))),s[1]||(s[1]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"F"),i("mo",null,"="),i("msup",null,[i("mn",null,"10"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mo",null,"−"),i("mn",null,"4")])])])],-1))]),s[3]||(s[3]=a(", the amplitude is a Lorentzian."))]),s[51]||(s[51]=n(`
    julia
    fixed = (α => 1, ω0 => 1.0, γ => 0.005, F => 0.0001)   # fixed parameters
+varied = ω => range(0.95, 1.05, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
+
+plot(result, "sqrt(u1^2 + v1^2)")

    ',2)),i("p",null,[s[6]||(s[6]=a("To find the fluctuation on the top of the steady state one often employs a ")),s[7]||(s[7]=i("a",{href:"https://en.wikipedia.org/wiki/Linear_dynamical_system",target:"_blank",rel:"noreferrer"},"Bogoliubov-de Gennes analyses",-1)),s[8]||(s[8]=a(". Here, we compute the eigenvalues ")),i("mjx-container",y,[(e(),t("svg",T,s[4]||(s[4]=[n('',1)]))),s[5]||(s[5]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"λ"),i("mi",null,"k")])])],-1))]),s[9]||(s[9]=a(' of the Jacobian matrix at the steady state. The imaginary part of the eigenvalues gives characteristic frequencies of the "quasi-particle excitations". The real part gives the lifetime of these excitations.'))]),s[52]||(s[52]=n(`

    One can plot the eigenvalues as follows

    julia
    plot(
+    plot_eigenvalues(result, branch=1),
+    plot_eigenvalues(result, branch=1, type=:real, ylims=(-0.003, 0)),
+)

    We find a single pair of complex conjugate eigenvalues linearly changing with the driving frequency. Both real parts are negative, indicating stability.

    As discussed in background section on linear response, the excitation manifest itself as a lorentenzian peak in a power spectral density (PSD) measurement. The PSD can be plotted using plot_linear_response:

    julia
    plot_linear_response(result, x, Ω_range=range(0.95, 1.05, 300), branch=1, logscale=true)

    ',7)),i("p",null,[s[16]||(s[16]=a("The response has a peak at ")),i("mjx-container",u,[(e(),t("svg",c,s[10]||(s[10]=[n('',1)]))),s[11]||(s[11]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"ω"),i("mn",null,"0")])])],-1))]),s[17]||(s[17]=a(", irrespective of the driving frequency ")),i("mjx-container",x,[(e(),t("svg",F,s[12]||(s[12]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[13]||(s[13]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω")])],-1))]),s[18]||(s[18]=a(". Indeed, the eigenvalues shown before where plotted in the rotating frame at the frequency of the drive ")),i("mjx-container",w,[(e(),t("svg",C,s[14]||(s[14]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[15]||(s[15]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω")])],-1))]),s[19]||(s[19]=a(". Hence, the imaginary part of eigenvalues shows the frequency (energy) needed to excite the system at it natural frequency (The frequency its want to be excited at.)"))]),i("p",null,[s[22]||(s[22]=a('Note the slight "bending" of the noise peak with ')),i("mjx-container",v,[(e(),t("svg",f,s[20]||(s[20]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),s[21]||(s[21]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω")])],-1))]),s[23]||(s[23]=a(" - this is given by the failure of the first-order calculation to capture response far-detuned from the drive frequency."))]),s[53]||(s[53]=i("h3",{id:"Nonlinear-regime",tabindex:"-1"},[a("Nonlinear regime "),i("a",{class:"header-anchor",href:"#Nonlinear-regime","aria-label":'Permalink to "Nonlinear regime {#Nonlinear-regime}"'},"​")],-1)),i("p",null,[s[26]||(s[26]=a("For strong driving, matters get more complicated. Let us now use a drive ")),i("mjx-container",b,[(e(),t("svg",B,s[24]||(s[24]=[n('',1)]))),s[25]||(s[25]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"F"),i("mo",null,"="),i("mn",null,"2"),i("mo",null,"∗"),i("msup",null,[i("mn",null,"10"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mo",null,"−"),i("mn",null,"3")])])])],-1))]),s[27]||(s[27]=a(" :"))]),s[54]||(s[54]=n(`
    julia
    fixed = (α => 1, ω0 => 1.0, γ => 0.005, F => 0.002)   # fixed parameters
+varied = ω => range(0.95, 1.05, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
+
+plot(result, x="ω", y="sqrt(u1^2 + v1^2)");

    The amplitude is the well-known Duffing curve. Let's look at the eigenvalues of the two stable branches, 1 and 2.

    julia
    plot(
+    plot_eigenvalues(result, branch=1),
+    plot_eigenvalues(result, branch=1, type=:real, ylims=(-0.003, 0)),
+    plot_eigenvalues(result, branch=2),
+    plot_eigenvalues(result, branch=2, type=:real, ylims=(-0.003, 0)),
+)

    ',5)),i("p",null,[s[30]||(s[30]=a("Again every branch gives a single pair of complex conjugate eigenvalues. However, for branch 1, the characteristic frequencies due not change linearly with the driving frequency around ")),i("mjx-container",D,[(e(),t("svg",H,s[28]||(s[28]=[n('',1)]))),s[29]||(s[29]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω"),i("mo",null,"="),i("msub",null,[i("mi",null,"ω"),i("mn",null,"0")])])],-1))]),s[31]||(s[31]=a(". This is a sign of steady state becoming nonlinear at large amplitudes."))]),s[55]||(s[55]=n(`

    The same can be seen in the PSD:

    julia
    plot(
+  plot_linear_response(result, x, branch=1, Ω_range=range(0.95,1.1,300), logscale=true),
+  plot_linear_response(result, x, branch=2, Ω_range=range(0.9,1.1,300), logscale=true),
+    size=(600, 250), margin=3mm
+)

    In branch 1 the linear response to white noise shows more than one peak. This is a distinctly nonlinear phenomenon, indicitive if the squeezing of the steady state. Branch 2 is again quasi-linear, which stems from its low amplitude.

    ',4)),i("p",null,[s[36]||(s[36]=a("Following ")),s[37]||(s[37]=i("a",{href:"https://doi.org/10.1103/PhysRevX.10.021066",target:"_blank",rel:"noreferrer"},"Huber et al.",-1)),s[38]||(s[38]=a(", we may also fix ")),i("mjx-container",A,[(e(),t("svg",M,s[32]||(s[32]=[n('',1)]))),s[33]||(s[33]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"ω"),i("mo",null,"="),i("msub",null,[i("mi",null,"ω"),i("mn",null,"0")])])],-1))]),s[39]||(s[39]=a(" and plot the linear response as a function of ")),i("mjx-container",L,[(e(),t("svg",j,s[34]||(s[34]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D439",d:"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z",style:{"stroke-width":"3"}})])])],-1)]))),s[35]||(s[35]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"F")])],-1))]),s[40]||(s[40]=a(". The response turns out to be single-valued over a large range of driving strengths. Using a log scale for the x-axis:"))]),s[56]||(s[56]=n(`
    julia
    fixed = (α => 1., ω0 => 1.0, γ => 1e-2, ω => 1)   # fixed parameters
+swept = F => 10 .^ range(-6, -1, 200)           # range of parameter values
+result = get_steady_states(harmonic_eq, swept, fixed)
+
+plot(
+  plot(result, "sqrt(u1^2 + v1^2)", xscale=:log),
+  plot_linear_response(result, x, branch=1, Ω_range=range(0.9,1.1,300), logscale=true, xscale=:log),
+  size=(600, 250), margin=3mm
+)

    ',2)),i("p",null,[s[45]||(s[45]=a("We see that for low ")),i("mjx-container",V,[(e(),t("svg",q,s[41]||(s[41]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D439",d:"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z",style:{"stroke-width":"3"}})])])],-1)]))),s[42]||(s[42]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"F")])],-1))]),s[46]||(s[46]=a(", quasi-linear behaviour with a single Lorentzian response occurs, while for larger ")),i("mjx-container",Z,[(e(),t("svg",z,s[43]||(s[43]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D439",d:"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z",style:{"stroke-width":"3"}})])])],-1)]))),s[44]||(s[44]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"F")])],-1))]),s[47]||(s[47]=a(", two peaks form in the noise response. The two peaks are strongly unequal in magnitude, which is an example of internal squeezing (See supplemental material of ")),s[48]||(s[48]=i("a",{href:"https://doi.org/10.1103/PhysRevX.10.021066",target:"_blank",rel:"noreferrer"},"Huber et al.",-1)),s[49]||(s[49]=a(")."))])])}const W=l(Q,[["render",S]]);export{O as __pageData,W as default}; diff --git a/v0.10.7/assets/tutorials_steady_states.md.DEM1FzKu.js b/v0.10.7/assets/tutorials_steady_states.md.DEM1FzKu.js new file mode 100644 index 00000000..bcbd1d50 --- /dev/null +++ b/v0.10.7/assets/tutorials_steady_states.md.DEM1FzKu.js @@ -0,0 +1,52 @@ +import{_ as l,c as T,j as t,a as Q,a4 as e,o as s}from"./chunks/framework.D3a-l5sU.js";const n="/HarmonicBalance.jl/v0.10.7/assets/okkxwnb.B07IzMp6.png",i="/HarmonicBalance.jl/v0.10.7/assets/qemjqtj.CY3KP9Dg.png",o="/HarmonicBalance.jl/v0.10.7/assets/ucpiacu.UTcoxLl5.png",r="/HarmonicBalance.jl/v0.10.7/assets/nmhmsgf.BNXvpC22.png",N1=JSON.parse('{"title":"Finding the staedy states of a Duffing 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For the Duffing resonator, we can well try to only consider the drive frequency ")),t("mjx-container",H,[(s(),T("svg",u,a[6]||(a[6]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[7]||(a[7]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[9]||(a[9]=Q(". To implement this, we use the ")),a[10]||(a[10]=t("em",null,"harmonic ansatz",-1))]),t("mjx-container",c,[(s(),T("svg",k,a[11]||(a[11]=[e('',1)]))),a[12]||(a[12]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"U"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"V"),t("mi",null,"sin"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mstyle",{scriptlevel:"0"},[t("mspace",{width:"0.167em"})]),t("mo",null,",")])],-1))]),t("p",null,[a[19]||(a[19]=Q("which constraints the spectrum of ")),t("mjx-container",w,[(s(),T("svg",x,a[13]||(a[13]=[e('',1)]))),a[14]||(a[14]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[20]||(a[20]=Q(" to a single harmonic. Fixing the quadratures ")),t("mjx-container",y,[(s(),T("svg",f,a[15]||(a[15]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D448",d:"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z",style:{"stroke-width":"3"}})])])],-1)]))),a[16]||(a[16]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"U")])],-1))]),a[21]||(a[21]=Q(" and ")),t("mjx-container",V,[(s(),T("svg",L,a[17]||(a[17]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D449",d:"M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z",style:{"stroke-width":"3"}})])])],-1)]))),a[18]||(a[18]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"V")])],-1))]),a[22]||(a[22]=Q(" to be constant then reduces the differential equation to two coupled cubic polynomial equations (for more details on this step, see the appendices in the ")),a[23]||(a[23]=t("a",{href:"https://scipost.org/SciPostPhysCodeb.6",target:"_blank",rel:"noreferrer"},"white paper",-1)),a[24]||(a[24]=Q("). Finding the roots of coupled polynomials is in general very hard. We here apply the method of homotopy continuation, as implemented in ")),a[25]||(a[25]=t("a",{href:"https://www.juliahomotopycontinuation.org/",target:"_blank",rel:"noreferrer"},"HomotopyContinuation.jl",-1)),a[26]||(a[26]=Q(" which is guaranteed to find the complete set of roots."))]),a[163]||(a[163]=e(`

    First we need to declare the symbolic variables (the excellent Symbolics.jl is used here).

    julia
    using HarmonicBalance
+@variables α ω ω0 F γ t x(t) # declare constant variables and a function x(t)

    Next, we have to input the equations of motion. This will be stored as a DifferentialEquation. The input needs to specify that only x is a mathematical variable, the other symbols are parameters:

    julia
    diff_eq = DifferentialEquation(d(x,t,2) + ω0^2*x + α*x^3 + γ*d(x,t) ~ F*cos(ω*t), x)
    System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ;   
+
+Differential(t)(Differential(t)(x(t))) + Differential(t)(x(t))*γ + x(t)*(ω0^2) + (x(t)^3)*α ~ F*cos(t*ω)

    One harmonic ​

    `,6)),t("p",null,[a[29]||(a[29]=Q("The harmonic ansatz needs to be specified now – we expand ")),a[30]||(a[30]=t("code",null,"x",-1)),a[31]||(a[31]=Q(" in a single frequency ")),t("mjx-container",M,[(s(),T("svg",v,a[27]||(a[27]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[28]||(a[28]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[32]||(a[32]=Q("."))]),a[164]||(a[164]=e('
    julia
    add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)
    ',1)),t("p",null,[a[37]||(a[37]=Q("The object ")),a[38]||(a[38]=t("code",null,"diff_eq",-1)),a[39]||(a[39]=Q(" now contains all the necessary information to convert the differential equation to the algebraic ")),a[40]||(a[40]=t("em",null,"harmonic equations",-1)),a[41]||(a[41]=Q(" (coupled polynomials in ")),t("mjx-container",b,[(s(),T("svg",Z,a[33]||(a[33]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D448",d:"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z",style:{"stroke-width":"3"}})])])],-1)]))),a[34]||(a[34]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"U")])],-1))]),a[42]||(a[42]=Q(" and ")),t("mjx-container",E,[(s(),T("svg",D,a[35]||(a[35]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D449",d:"M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z",style:{"stroke-width":"3"}})])])],-1)]))),a[36]||(a[36]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"V")])],-1))]),a[43]||(a[43]=Q(")."))]),a[165]||(a[165]=e(`
    julia
    harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω0, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ F
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0

    The variables u1 and v1 were declared automatically to construct the harmonic ansatz. The slow time variable T describes variation of the quadratures on timescales much slower than ω. For a steady state, all derivatives w.r.t T vanish, leaving only algebraic equations to be solved.

    We are ready to start plugging in numbers! Let us find steady states by solving harmonic_eq for numerical parameters. Homotopy continuation is especially suited to solving over a range of parameter values. Here we will solve over a range of driving frequencies ω – these are stored as Pairs{Sym, Vector{Float64}}:

    julia
    varied = ω => range(0.9, 1.2, 100); # range of parameter values
    ω => 0.9:0.0030303030303030303:1.2

    The other parameters we be fixed – these are declared as Pairs{Sym, Float64} pairs:

    julia
    fixed = (α => 1., ω0 => 1.0, F => 0.01, γ => 0.01); # fixed parameters
    (α => 1.0, ω0 => 1.0, F => 0.01, γ => 0.01)

    Now everything is ready to crank the handle. get_steady_states solves our harmonic_eq using the varied and fixed parameters:

    julia
    result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 100 parameter points
+
+Solution branches:   3
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels
    `,12)),t("p",null,[a[48]||(a[48]=Q("The algorithm has found 3 solution branches in total (out of the ")),a[49]||(a[49]=t("a",{href:"https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem",target:"_blank",rel:"noreferrer"},"hypothetically admissible",-1)),a[50]||(a[50]=Q()),t("mjx-container",C,[(s(),T("svg",j,a[44]||(a[44]=[e('',1)]))),a[45]||(a[45]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mn",null,"3"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2")])]),t("mo",null,"="),t("mn",null,"9")])],-1))]),a[51]||(a[51]=Q("). All of these are real – and thefore physically observable – for at least some values of ")),t("mjx-container",F,[(s(),T("svg",B,a[46]||(a[46]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[47]||(a[47]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[52]||(a[52]=Q('. Only 2 branches are stable under infinitesimal perturbations. The "Classes" are boolean labels classifying each solution point, which may be used to select results for plotting.'))]),t("p",null,[a[57]||(a[57]=Q("We now want to visualize the results. Here we plot the solution amplitude, ")),t("mjx-container",A,[(s(),T("svg",q,a[53]||(a[53]=[e('',1)]))),a[54]||(a[54]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msqrt",null,[t("msup",null,[t("mi",null,"U"),t("mn",null,"2")]),t("mo",null,"+"),t("msup",null,[t("mi",null,"V"),t("mn",null,"2")])])])],-1))]),a[58]||(a[58]=Q(" against the drive frequency ")),t("mjx-container",z,[(s(),T("svg",S,a[55]||(a[55]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[56]||(a[56]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[59]||(a[59]=Q(":"))]),a[166]||(a[166]=e('
    julia
    plot(result, "sqrt(u1^2 + v1^2)")

    This is the expected response curve for the Duffing equation.

    Using multiple harmonics ​

    ',4)),t("p",null,[a[64]||(a[64]=Q("In the above section, we truncated the Fourier space to a single harmonic ")),t("mjx-container",O,[(s(),T("svg",J,a[60]||(a[60]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[61]||(a[61]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[65]||(a[65]=Q(" – the oscillator was assumed to only oscillate at the drive frequency. However, the Duffing oscillator can exhibit a rich spectrum of harmonics. We can obtain some intuition by treating ")),t("mjx-container",G,[(s(),T("svg",N,a[62]||(a[62]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),a[63]||(a[63]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"α")])],-1))]),a[66]||(a[66]=Q(" perturbatively in the equation of motion, i.e., by solving"))]),t("mjx-container",R,[(s(),T("svg",P,a[67]||(a[67]=[e('',1)]))),a[68]||(a[68]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"γ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"ω"),t("mn",null,"0"),t("mn",null,"2")]),t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"ϵ"),t("mi",null,"α"),t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mn",null,"3")]),t("mo",null,"="),t("mi",null,"F"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),t("p",null,[a[77]||(a[77]=Q("for small ")),t("mjx-container",X,[(s(),T("svg",U,a[69]||(a[69]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D716",d:"M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z",style:{"stroke-width":"3"}})])])],-1)]))),a[70]||(a[70]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ϵ")])],-1))]),a[78]||(a[78]=Q(". To zeroth order, the response of the system is ")),t("mjx-container",W,[(s(),T("svg",I,a[71]||(a[71]=[e('',1)]))),a[72]||(a[72]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mn",null,"0")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mi",null,"X"),t("mn",null,"0")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("msub",null,[t("mi",null,"ϕ"),t("mn",null,"0")]),t("mo",{stretchy:"false"},")")])],-1))]),a[79]||(a[79]=Q(". Expanding ")),t("mjx-container",$,[(s(),T("svg",K,a[73]||(a[73]=[e('',1)]))),a[74]||(a[74]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mi",null,"x"),t("mn",null,"0")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"ϵ"),t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[80]||(a[80]=Q(", we find that the perturbation ")),t("mjx-container",Y,[(s(),T("svg",_,a[75]||(a[75]=[e('',1)]))),a[76]||(a[76]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[81]||(a[81]=Q(" satisfies to first order"))]),t("mjx-container",t1,[(s(),T("svg",a1,a[82]||(a[82]=[e('',1)]))),a[83]||(a[83]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"γ"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"INNER"},[t("mo",{"data-mjx-texclass":"OPEN"},"["),t("msubsup",null,[t("mi",null,"ω"),t("mn",null,"0"),t("mn",null,"2")]),t("mo",null,"+"),t("mfrac",null,[t("mrow",null,[t("mn",null,"3"),t("mi",null,"α"),t("msubsup",null,[t("mi",null,"X"),t("mn",null,"0"),t("mn",null,"2")])]),t("mn",null,"4")]),t("mo",{"data-mjx-texclass":"CLOSE"},"]")]),t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mo",null,"−"),t("mfrac",null,[t("mrow",null,[t("mi",null,"α"),t("msubsup",null,[t("mi",null,"X"),t("mn",null,"0"),t("mn",null,"3")])]),t("mn",null,"4")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"3"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("mn",null,"3"),t("msub",null,[t("mi",null,"ϕ"),t("mn",null,"0")]),t("mo",{stretchy:"false"},")"),t("mstyle",{scriptlevel:"0"},[t("mspace",{width:"0.167em"})]),t("mo",null,",")])],-1))]),t("p",null,[a[90]||(a[90]=Q("which gives a response of the form ")),t("mjx-container",Q1,[(s(),T("svg",T1,a[84]||(a[84]=[e('',1)]))),a[85]||(a[85]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mi",null,"X"),t("mn",null,"1")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"3"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("msub",null,[t("mi",null,"ϕ"),t("mn",null,"1")]),t("mo",{stretchy:"false"},")")])],-1))]),a[91]||(a[91]=Q(". Clearly, the oscillator now responds not only at frequency ")),t("mjx-container",s1,[(s(),T("svg",e1,a[86]||(a[86]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[87]||(a[87]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[92]||(a[92]=Q(", but also at ")),t("mjx-container",l1,[(s(),T("svg",n1,a[88]||(a[88]=[e('',1)]))),a[89]||(a[89]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω")])],-1))]),a[93]||(a[93]=Q("! This effect is known as ")),a[94]||(a[94]=t("a",{href:"https://en.wikipedia.org/wiki/High_harmonic_generation",target:"_blank",rel:"noreferrer"},[t("em",null,"high harmonic generation")],-1)),a[95]||(a[95]=Q(" or more generally ")),a[96]||(a[96]=t("em",null,"frequency conversion",-1)),a[97]||(a[97]=Q(". By continuing the procedure to higher orders, we eventually obtain an infinity of harmonics present in the response. In general, there is no analytical solution to such problems."))]),t("p",null,[a[102]||(a[102]=Q("We argued that frequency conversion takes place, to first order from ")),t("mjx-container",i1,[(s(),T("svg",o1,a[98]||(a[98]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[99]||(a[99]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[103]||(a[103]=Q(" to ")),t("mjx-container",r1,[(s(),T("svg",d1,a[100]||(a[100]=[e('',1)]))),a[101]||(a[101]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω")])],-1))]),a[104]||(a[104]=Q(". We can reflect this process by using a extended harmonic ansatz:"))]),t("mjx-container",m1,[(s(),T("svg",h1,a[105]||(a[105]=[e('',1)]))),a[106]||(a[106]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mi",null,"U"),t("mn",null,"1")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msub",null,[t("mi",null,"V"),t("mn",null,"1")]),t("mi",null,"sin"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msub",null,[t("mi",null,"U"),t("mn",null,"2")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"3"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msub",null,[t("mi",null,"V"),t("mn",null,"2")]),t("mi",null,"sin"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"3"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mstyle",{scriptlevel:"0"},[t("mspace",{width:"0.167em"})]),t("mo",null,".")])],-1))]),t("p",null,[a[111]||(a[111]=Q("Note that this is not a perturbative treatment! The harmonics ")),t("mjx-container",p1,[(s(),T("svg",g1,a[107]||(a[107]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[108]||(a[108]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[112]||(a[112]=Q(" and ")),t("mjx-container",H1,[(s(),T("svg",u1,a[109]||(a[109]=[e('',1)]))),a[110]||(a[110]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω")])],-1))]),a[113]||(a[113]=Q(" are on the same footing here. This is implemented as"))]),a[167]||(a[167]=e(`
    julia
    add_harmonic!(diff_eq, x, [ω, 3ω]) # specify the two-harmonics ansatz
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 4 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T)
+Parameters: ω, ω0, γ, α, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt) + u2(T)*cos(3ωt) + v2(T)*sin(3ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*u2(T)*α + (3//2)*u1(T)*(v2(T)^2)*α + (3//2)*u1(T)*v2(T)*v1(T)*α + (3//4)*u1(T)*(v1(T)^2)*α + (3//2)*u1(T)*(u2(T)^2)*α - (3//4)*(v1(T)^2)*u2(T)*α ~ F
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v2(T)*α + (3//4)*(u1(T)^2)*v1(T)*α - (3//2)*u1(T)*v1(T)*u2(T)*α + (3//2)*(v2(T)^2)*v1(T)*α - (3//4)*v2(T)*(v1(T)^2)*α + (3//4)*(v1(T)^3)*α + (3//2)*v1(T)*(u2(T)^2)*α ~ 0//1
+
+Differential(T)(u2(T))*γ + (6//1)*Differential(T)(v2(T))*ω + (3//1)*v2(T)*γ*ω - (9//1)*u2(T)*(ω^2) + u2(T)*(ω0^2) + (1//4)*(u1(T)^3)*α + (3//2)*(u1(T)^2)*u2(T)*α - (3//4)*u1(T)*(v1(T)^2)*α + (3//4)*(v2(T)^2)*u2(T)*α + (3//2)*(v1(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0//1
+
+-(6//1)*Differential(T)(u2(T))*ω + Differential(T)(v2(T))*γ - (9//1)*v2(T)*(ω^2) + v2(T)*(ω0^2) - (3//1)*u2(T)*γ*ω + (3//2)*(u1(T)^2)*v2(T)*α + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v2(T)^3)*α + (3//2)*v2(T)*(v1(T)^2)*α + (3//4)*v2(T)*(u2(T)^2)*α - (1//4)*(v1(T)^3)*α ~ 0//1
    `,2)),t("p",null,[a[116]||(a[116]=Q("The variables ")),a[117]||(a[117]=t("code",null,"u1",-1)),a[118]||(a[118]=Q(", ")),a[119]||(a[119]=t("code",null,"v1",-1)),a[120]||(a[120]=Q(" now encode ")),a[121]||(a[121]=t("code",null,"ω",-1)),a[122]||(a[122]=Q(" and ")),a[123]||(a[123]=t("code",null,"u2",-1)),a[124]||(a[124]=Q(", ")),a[125]||(a[125]=t("code",null,"v2",-1)),a[126]||(a[126]=Q(" encode ")),a[127]||(a[127]=t("code",null,"3ω",-1)),a[128]||(a[128]=Q(". We see this system is much harder to solve as we now have 4 harmonic variables, resulting in 4 coupled cubic equations. A maximum of ")),t("mjx-container",c1,[(s(),T("svg",k1,a[114]||(a[114]=[e('',1)]))),a[115]||(a[115]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mn",null,"3"),t("mn",null,"4")]),t("mo",null,"="),t("mn",null,"81")])],-1))]),a[129]||(a[129]=Q(" solutions ")),a[130]||(a[130]=t("a",{href:"https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem",target:"_blank",rel:"noreferrer"},"may appear",-1)),a[131]||(a[131]=Q("!"))]),a[168]||(a[168]=e(`
    julia
    result = get_steady_states(harmonic_eq, varied, fixed)
+plot(result, "sqrt(u1^2 + v1^2)")

    ',2)),t("p",null,[a[136]||(a[136]=Q("For the above parameters (where a perturbative treatment would have been reasonable), the principal response at ")),t("mjx-container",w1,[(s(),T("svg",x1,a[132]||(a[132]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[133]||(a[133]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[137]||(a[137]=Q(" looks rather similar, with a much smaller upconverted component appearing at ")),t("mjx-container",y1,[(s(),T("svg",f1,a[134]||(a[134]=[e('',1)]))),a[135]||(a[135]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω")])],-1))]),a[138]||(a[138]=Q(":"))]),a[169]||(a[169]=e(`
    julia
    p1=plot(result, "sqrt(u1^2 + v1^2)", legend=false)
+p2=plot(result, "sqrt(u2^2 + v2^2)")
+plot(p1, p2)

    ',2)),t("p",null,[a[143]||(a[143]=Q("The non-perturbative nature of the ansatz allows us to capture some behaviour which is ")),a[144]||(a[144]=t("em",null,"not",-1)),a[145]||(a[145]=Q(" a mere extension of the usual single-harmonic Duffing response. Suppose we drive a strongly nonlinear resonator at frequency ")),t("mjx-container",V1,[(s(),T("svg",L1,a[139]||(a[139]=[e('',1)]))),a[140]||(a[140]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω"),t("mo",null,"≅"),t("msub",null,[t("mi",null,"ω"),t("mn",null,"0")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mn",null,"3")])],-1))]),a[146]||(a[146]=Q(". Such a drive is far out of resonance, however, the upconverted harmonic ")),t("mjx-container",M1,[(s(),T("svg",v1,a[141]||(a[141]=[e('',1)]))),a[142]||(a[142]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω"),t("mo",null,"="),t("msub",null,[t("mi",null,"ω"),t("mn",null,"0")])])],-1))]),a[147]||(a[147]=Q(" is not and may play an important role! Let us try this out:"))]),a[170]||(a[170]=e(`
    julia
    fixed = (α => 10., ω0 => 3, F => 5, γ=>0.01)   # fixed parameters
+varied = ω => range(0.9, 1.4, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 100 parameter points
+
+Solution branches:   9
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels
    `,2)),t("p",null,[a[152]||(a[152]=Q("Although 9 branches were found in total, only 3 remain physical (real-valued). Let us visualise the amplitudes corresponding to the two harmonics, ")),t("mjx-container",b1,[(s(),T("svg",Z1,a[148]||(a[148]=[e('',1)]))),a[149]||(a[149]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msqrt",null,[t("msubsup",null,[t("mi",null,"U"),t("mn",null,"1"),t("mn",null,"2")]),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"V"),t("mn",null,"1"),t("mn",null,"2")])])])],-1))]),a[153]||(a[153]=Q(" and ")),t("mjx-container",E1,[(s(),T("svg",D1,a[150]||(a[150]=[e('',1)]))),a[151]||(a[151]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msqrt",null,[t("msubsup",null,[t("mi",null,"U"),t("mn",null,"2"),t("mn",null,"2")]),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"V"),t("mn",null,"2"),t("mn",null,"2")])])])],-1))]),a[154]||(a[154]=Q(" :"))]),a[171]||(a[171]=e(`
    julia
    p1 = plot(result, "sqrt(u1^2 + v1^2)", legend=false)
+p2 = plot(result, "sqrt(u2^2 + v2^2)")
+plot(p1, p2)

    ',2)),t("p",null,[a[159]||(a[159]=Q("The contributions of ")),t("mjx-container",C1,[(s(),T("svg",j1,a[155]||(a[155]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[156]||(a[156]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[160]||(a[160]=Q(" and ")),t("mjx-container",F1,[(s(),T("svg",B1,a[157]||(a[157]=[e('',1)]))),a[158]||(a[158]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω")])],-1))]),a[161]||(a[161]=Q(" are now comparable and the system shows some fairly complex behaviour! This demonstrates how an exact solution within an extended Fourier subspace goes beyond a perturbative treatment."))])])}const R1=l(d,[["render",A1]]);export{N1 as __pageData,R1 as default}; diff --git a/v0.10.7/assets/tutorials_steady_states.md.DEM1FzKu.lean.js b/v0.10.7/assets/tutorials_steady_states.md.DEM1FzKu.lean.js new file mode 100644 index 00000000..bcbd1d50 --- /dev/null +++ b/v0.10.7/assets/tutorials_steady_states.md.DEM1FzKu.lean.js @@ -0,0 +1,52 @@ +import{_ as l,c as T,j as t,a as Q,a4 as e,o as s}from"./chunks/framework.D3a-l5sU.js";const n="/HarmonicBalance.jl/v0.10.7/assets/okkxwnb.B07IzMp6.png",i="/HarmonicBalance.jl/v0.10.7/assets/qemjqtj.CY3KP9Dg.png",o="/HarmonicBalance.jl/v0.10.7/assets/ucpiacu.UTcoxLl5.png",r="/HarmonicBalance.jl/v0.10.7/assets/nmhmsgf.BNXvpC22.png",N1=JSON.parse('{"title":"Finding the staedy states of a Duffing 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{#Duffing}"'},"​")],-1)),t("p",null,[a[2]||(a[2]=Q("Here we show the workflow of HarmonicBalance.jl on a simple example - the driven Duffing oscillator. The equation of motion for the displacement ")),t("mjx-container",m,[(s(),T("svg",h,a[0]||(a[0]=[e('',1)]))),a[1]||(a[1]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[3]||(a[3]=Q(" reads"))]),t("mjx-container",p,[(s(),T("svg",g,a[4]||(a[4]=[e('',1)]))),a[5]||(a[5]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("munder",null,[t("mrow",{"data-mjx-texclass":"OP"},[t("munder",null,[t("mrow",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"γ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"ω"),t("mn",null,"0"),t("mn",null,"2")]),t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")]),t("mo",null,"⏟")])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"damped harmonic oscillator")])]),t("mo",null,"+"),t("munder",null,[t("mrow",{"data-mjx-texclass":"OP"},[t("munder",null,[t("mrow",null,[t("mi",null,"α"),t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mn",null,"3")])]),t("mo",null,"⏟")])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"Duffing coefficient")])]),t("mo",null,"="),t("munder",null,[t("mrow",{"data-mjx-texclass":"OP"},[t("munder",null,[t("mrow",null,[t("mi",null,"F"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")]),t("mo",null,"⏟")])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mtext",null,"periodic drive")])])])],-1))]),t("p",null,[a[8]||(a[8]=Q("In general, there is no analytical solution to the differential equation. Fortunately, some harmonics are more important than others. By truncating the infinite-dimensional Fourier space to a set of judiciously chosen harmonics, we may obtain a soluble system. For the Duffing resonator, we can well try to only consider the drive frequency ")),t("mjx-container",H,[(s(),T("svg",u,a[6]||(a[6]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[7]||(a[7]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[9]||(a[9]=Q(". To implement this, we use the ")),a[10]||(a[10]=t("em",null,"harmonic ansatz",-1))]),t("mjx-container",c,[(s(),T("svg",k,a[11]||(a[11]=[e('',1)]))),a[12]||(a[12]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"U"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"V"),t("mi",null,"sin"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mstyle",{scriptlevel:"0"},[t("mspace",{width:"0.167em"})]),t("mo",null,",")])],-1))]),t("p",null,[a[19]||(a[19]=Q("which constraints the spectrum of ")),t("mjx-container",w,[(s(),T("svg",x,a[13]||(a[13]=[e('',1)]))),a[14]||(a[14]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[20]||(a[20]=Q(" to a single harmonic. Fixing the quadratures ")),t("mjx-container",y,[(s(),T("svg",f,a[15]||(a[15]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D448",d:"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z",style:{"stroke-width":"3"}})])])],-1)]))),a[16]||(a[16]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"U")])],-1))]),a[21]||(a[21]=Q(" and ")),t("mjx-container",V,[(s(),T("svg",L,a[17]||(a[17]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D449",d:"M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z",style:{"stroke-width":"3"}})])])],-1)]))),a[18]||(a[18]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"V")])],-1))]),a[22]||(a[22]=Q(" to be constant then reduces the differential equation to two coupled cubic polynomial equations (for more details on this step, see the appendices in the ")),a[23]||(a[23]=t("a",{href:"https://scipost.org/SciPostPhysCodeb.6",target:"_blank",rel:"noreferrer"},"white paper",-1)),a[24]||(a[24]=Q("). Finding the roots of coupled polynomials is in general very hard. We here apply the method of homotopy continuation, as implemented in ")),a[25]||(a[25]=t("a",{href:"https://www.juliahomotopycontinuation.org/",target:"_blank",rel:"noreferrer"},"HomotopyContinuation.jl",-1)),a[26]||(a[26]=Q(" which is guaranteed to find the complete set of roots."))]),a[163]||(a[163]=e(`

    First we need to declare the symbolic variables (the excellent Symbolics.jl is used here).

    julia
    using HarmonicBalance
+@variables α ω ω0 F γ t x(t) # declare constant variables and a function x(t)

    Next, we have to input the equations of motion. This will be stored as a DifferentialEquation. The input needs to specify that only x is a mathematical variable, the other symbols are parameters:

    julia
    diff_eq = DifferentialEquation(d(x,t,2) + ω0^2*x + α*x^3 + γ*d(x,t) ~ F*cos(ω*t), x)
    System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ;   
+
+Differential(t)(Differential(t)(x(t))) + Differential(t)(x(t))*γ + x(t)*(ω0^2) + (x(t)^3)*α ~ F*cos(t*ω)

    One harmonic ​

    `,6)),t("p",null,[a[29]||(a[29]=Q("The harmonic ansatz needs to be specified now – we expand ")),a[30]||(a[30]=t("code",null,"x",-1)),a[31]||(a[31]=Q(" in a single frequency ")),t("mjx-container",M,[(s(),T("svg",v,a[27]||(a[27]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[28]||(a[28]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[32]||(a[32]=Q("."))]),a[164]||(a[164]=e('
    julia
    add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)
    ',1)),t("p",null,[a[37]||(a[37]=Q("The object ")),a[38]||(a[38]=t("code",null,"diff_eq",-1)),a[39]||(a[39]=Q(" now contains all the necessary information to convert the differential equation to the algebraic ")),a[40]||(a[40]=t("em",null,"harmonic equations",-1)),a[41]||(a[41]=Q(" (coupled polynomials in ")),t("mjx-container",b,[(s(),T("svg",Z,a[33]||(a[33]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D448",d:"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z",style:{"stroke-width":"3"}})])])],-1)]))),a[34]||(a[34]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"U")])],-1))]),a[42]||(a[42]=Q(" and ")),t("mjx-container",E,[(s(),T("svg",D,a[35]||(a[35]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D449",d:"M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z",style:{"stroke-width":"3"}})])])],-1)]))),a[36]||(a[36]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"V")])],-1))]),a[43]||(a[43]=Q(")."))]),a[165]||(a[165]=e(`
    julia
    harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω0, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ F
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0

    The variables u1 and v1 were declared automatically to construct the harmonic ansatz. The slow time variable T describes variation of the quadratures on timescales much slower than ω. For a steady state, all derivatives w.r.t T vanish, leaving only algebraic equations to be solved.

    We are ready to start plugging in numbers! Let us find steady states by solving harmonic_eq for numerical parameters. Homotopy continuation is especially suited to solving over a range of parameter values. Here we will solve over a range of driving frequencies ω – these are stored as Pairs{Sym, Vector{Float64}}:

    julia
    varied = ω => range(0.9, 1.2, 100); # range of parameter values
    ω => 0.9:0.0030303030303030303:1.2

    The other parameters we be fixed – these are declared as Pairs{Sym, Float64} pairs:

    julia
    fixed = (α => 1., ω0 => 1.0, F => 0.01, γ => 0.01); # fixed parameters
    (α => 1.0, ω0 => 1.0, F => 0.01, γ => 0.01)

    Now everything is ready to crank the handle. get_steady_states solves our harmonic_eq using the varied and fixed parameters:

    julia
    result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 100 parameter points
+
+Solution branches:   3
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels
    `,12)),t("p",null,[a[48]||(a[48]=Q("The algorithm has found 3 solution branches in total (out of the ")),a[49]||(a[49]=t("a",{href:"https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem",target:"_blank",rel:"noreferrer"},"hypothetically admissible",-1)),a[50]||(a[50]=Q()),t("mjx-container",C,[(s(),T("svg",j,a[44]||(a[44]=[e('',1)]))),a[45]||(a[45]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mn",null,"3"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"2")])]),t("mo",null,"="),t("mn",null,"9")])],-1))]),a[51]||(a[51]=Q("). All of these are real – and thefore physically observable – for at least some values of ")),t("mjx-container",F,[(s(),T("svg",B,a[46]||(a[46]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[47]||(a[47]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[52]||(a[52]=Q('. Only 2 branches are stable under infinitesimal perturbations. The "Classes" are boolean labels classifying each solution point, which may be used to select results for plotting.'))]),t("p",null,[a[57]||(a[57]=Q("We now want to visualize the results. Here we plot the solution amplitude, ")),t("mjx-container",A,[(s(),T("svg",q,a[53]||(a[53]=[e('',1)]))),a[54]||(a[54]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msqrt",null,[t("msup",null,[t("mi",null,"U"),t("mn",null,"2")]),t("mo",null,"+"),t("msup",null,[t("mi",null,"V"),t("mn",null,"2")])])])],-1))]),a[58]||(a[58]=Q(" against the drive frequency ")),t("mjx-container",z,[(s(),T("svg",S,a[55]||(a[55]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[56]||(a[56]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[59]||(a[59]=Q(":"))]),a[166]||(a[166]=e('
    julia
    plot(result, "sqrt(u1^2 + v1^2)")

    This is the expected response curve for the Duffing equation.

    Using multiple harmonics ​

    ',4)),t("p",null,[a[64]||(a[64]=Q("In the above section, we truncated the Fourier space to a single harmonic ")),t("mjx-container",O,[(s(),T("svg",J,a[60]||(a[60]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[61]||(a[61]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[65]||(a[65]=Q(" – the oscillator was assumed to only oscillate at the drive frequency. However, the Duffing oscillator can exhibit a rich spectrum of harmonics. We can obtain some intuition by treating ")),t("mjx-container",G,[(s(),T("svg",N,a[62]||(a[62]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),a[63]||(a[63]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"α")])],-1))]),a[66]||(a[66]=Q(" perturbatively in the equation of motion, i.e., by solving"))]),t("mjx-container",R,[(s(),T("svg",P,a[67]||(a[67]=[e('',1)]))),a[68]||(a[68]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"γ"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"ω"),t("mn",null,"0"),t("mn",null,"2")]),t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"ϵ"),t("mi",null,"α"),t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mn",null,"3")]),t("mo",null,"="),t("mi",null,"F"),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),t("p",null,[a[77]||(a[77]=Q("for small ")),t("mjx-container",X,[(s(),T("svg",U,a[69]||(a[69]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D716",d:"M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z",style:{"stroke-width":"3"}})])])],-1)]))),a[70]||(a[70]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ϵ")])],-1))]),a[78]||(a[78]=Q(". To zeroth order, the response of the system is ")),t("mjx-container",W,[(s(),T("svg",I,a[71]||(a[71]=[e('',1)]))),a[72]||(a[72]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mn",null,"0")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mi",null,"X"),t("mn",null,"0")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("msub",null,[t("mi",null,"ϕ"),t("mn",null,"0")]),t("mo",{stretchy:"false"},")")])],-1))]),a[79]||(a[79]=Q(". Expanding ")),t("mjx-container",$,[(s(),T("svg",K,a[73]||(a[73]=[e('',1)]))),a[74]||(a[74]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mi",null,"x"),t("mn",null,"0")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"ϵ"),t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[80]||(a[80]=Q(", we find that the perturbation ")),t("mjx-container",Y,[(s(),T("svg",_,a[75]||(a[75]=[e('',1)]))),a[76]||(a[76]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")")])],-1))]),a[81]||(a[81]=Q(" satisfies to first order"))]),t("mjx-container",t1,[(s(),T("svg",a1,a[82]||(a[82]=[e('',1)]))),a[83]||(a[83]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"¨")])]),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("mi",null,"γ"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mover",null,[t("mi",null,"x"),t("mo",null,"˙")])]),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"INNER"},[t("mo",{"data-mjx-texclass":"OPEN"},"["),t("msubsup",null,[t("mi",null,"ω"),t("mn",null,"0"),t("mn",null,"2")]),t("mo",null,"+"),t("mfrac",null,[t("mrow",null,[t("mn",null,"3"),t("mi",null,"α"),t("msubsup",null,[t("mi",null,"X"),t("mn",null,"0"),t("mn",null,"2")])]),t("mn",null,"4")]),t("mo",{"data-mjx-texclass":"CLOSE"},"]")]),t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mo",null,"−"),t("mfrac",null,[t("mrow",null,[t("mi",null,"α"),t("msubsup",null,[t("mi",null,"X"),t("mn",null,"0"),t("mn",null,"3")])]),t("mn",null,"4")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"3"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("mn",null,"3"),t("msub",null,[t("mi",null,"ϕ"),t("mn",null,"0")]),t("mo",{stretchy:"false"},")"),t("mstyle",{scriptlevel:"0"},[t("mspace",{width:"0.167em"})]),t("mo",null,",")])],-1))]),t("p",null,[a[90]||(a[90]=Q("which gives a response of the form ")),t("mjx-container",Q1,[(s(),T("svg",T1,a[84]||(a[84]=[e('',1)]))),a[85]||(a[85]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"x"),t("mn",null,"1")]),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mi",null,"X"),t("mn",null,"1")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"3"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",null,"+"),t("msub",null,[t("mi",null,"ϕ"),t("mn",null,"1")]),t("mo",{stretchy:"false"},")")])],-1))]),a[91]||(a[91]=Q(". Clearly, the oscillator now responds not only at frequency ")),t("mjx-container",s1,[(s(),T("svg",e1,a[86]||(a[86]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[87]||(a[87]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[92]||(a[92]=Q(", but also at ")),t("mjx-container",l1,[(s(),T("svg",n1,a[88]||(a[88]=[e('',1)]))),a[89]||(a[89]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω")])],-1))]),a[93]||(a[93]=Q("! This effect is known as ")),a[94]||(a[94]=t("a",{href:"https://en.wikipedia.org/wiki/High_harmonic_generation",target:"_blank",rel:"noreferrer"},[t("em",null,"high harmonic generation")],-1)),a[95]||(a[95]=Q(" or more generally ")),a[96]||(a[96]=t("em",null,"frequency conversion",-1)),a[97]||(a[97]=Q(". By continuing the procedure to higher orders, we eventually obtain an infinity of harmonics present in the response. In general, there is no analytical solution to such problems."))]),t("p",null,[a[102]||(a[102]=Q("We argued that frequency conversion takes place, to first order from ")),t("mjx-container",i1,[(s(),T("svg",o1,a[98]||(a[98]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[99]||(a[99]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[103]||(a[103]=Q(" to ")),t("mjx-container",r1,[(s(),T("svg",d1,a[100]||(a[100]=[e('',1)]))),a[101]||(a[101]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω")])],-1))]),a[104]||(a[104]=Q(". We can reflect this process by using a extended harmonic ansatz:"))]),t("mjx-container",m1,[(s(),T("svg",h1,a[105]||(a[105]=[e('',1)]))),a[106]||(a[106]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"x"),t("mo",{stretchy:"false"},"("),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("msub",null,[t("mi",null,"U"),t("mn",null,"1")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msub",null,[t("mi",null,"V"),t("mn",null,"1")]),t("mi",null,"sin"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msub",null,[t("mi",null,"U"),t("mn",null,"2")]),t("mi",null,"cos"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"3"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mo",null,"+"),t("msub",null,[t("mi",null,"V"),t("mn",null,"2")]),t("mi",null,"sin"),t("mo",{"data-mjx-texclass":"NONE"},"⁡"),t("mo",{stretchy:"false"},"("),t("mn",null,"3"),t("mi",null,"ω"),t("mi",null,"t"),t("mo",{stretchy:"false"},")"),t("mstyle",{scriptlevel:"0"},[t("mspace",{width:"0.167em"})]),t("mo",null,".")])],-1))]),t("p",null,[a[111]||(a[111]=Q("Note that this is not a perturbative treatment! The harmonics ")),t("mjx-container",p1,[(s(),T("svg",g1,a[107]||(a[107]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[108]||(a[108]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[112]||(a[112]=Q(" and ")),t("mjx-container",H1,[(s(),T("svg",u1,a[109]||(a[109]=[e('',1)]))),a[110]||(a[110]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω")])],-1))]),a[113]||(a[113]=Q(" are on the same footing here. This is implemented as"))]),a[167]||(a[167]=e(`
    julia
    add_harmonic!(diff_eq, x, [ω, 3ω]) # specify the two-harmonics ansatz
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 4 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T)
+Parameters: ω, ω0, γ, α, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt) + u2(T)*cos(3ωt) + v2(T)*sin(3ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*u2(T)*α + (3//2)*u1(T)*(v2(T)^2)*α + (3//2)*u1(T)*v2(T)*v1(T)*α + (3//4)*u1(T)*(v1(T)^2)*α + (3//2)*u1(T)*(u2(T)^2)*α - (3//4)*(v1(T)^2)*u2(T)*α ~ F
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v2(T)*α + (3//4)*(u1(T)^2)*v1(T)*α - (3//2)*u1(T)*v1(T)*u2(T)*α + (3//2)*(v2(T)^2)*v1(T)*α - (3//4)*v2(T)*(v1(T)^2)*α + (3//4)*(v1(T)^3)*α + (3//2)*v1(T)*(u2(T)^2)*α ~ 0//1
+
+Differential(T)(u2(T))*γ + (6//1)*Differential(T)(v2(T))*ω + (3//1)*v2(T)*γ*ω - (9//1)*u2(T)*(ω^2) + u2(T)*(ω0^2) + (1//4)*(u1(T)^3)*α + (3//2)*(u1(T)^2)*u2(T)*α - (3//4)*u1(T)*(v1(T)^2)*α + (3//4)*(v2(T)^2)*u2(T)*α + (3//2)*(v1(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0//1
+
+-(6//1)*Differential(T)(u2(T))*ω + Differential(T)(v2(T))*γ - (9//1)*v2(T)*(ω^2) + v2(T)*(ω0^2) - (3//1)*u2(T)*γ*ω + (3//2)*(u1(T)^2)*v2(T)*α + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v2(T)^3)*α + (3//2)*v2(T)*(v1(T)^2)*α + (3//4)*v2(T)*(u2(T)^2)*α - (1//4)*(v1(T)^3)*α ~ 0//1
    `,2)),t("p",null,[a[116]||(a[116]=Q("The variables ")),a[117]||(a[117]=t("code",null,"u1",-1)),a[118]||(a[118]=Q(", ")),a[119]||(a[119]=t("code",null,"v1",-1)),a[120]||(a[120]=Q(" now encode ")),a[121]||(a[121]=t("code",null,"ω",-1)),a[122]||(a[122]=Q(" and ")),a[123]||(a[123]=t("code",null,"u2",-1)),a[124]||(a[124]=Q(", ")),a[125]||(a[125]=t("code",null,"v2",-1)),a[126]||(a[126]=Q(" encode ")),a[127]||(a[127]=t("code",null,"3ω",-1)),a[128]||(a[128]=Q(". We see this system is much harder to solve as we now have 4 harmonic variables, resulting in 4 coupled cubic equations. A maximum of ")),t("mjx-container",c1,[(s(),T("svg",k1,a[114]||(a[114]=[e('',1)]))),a[115]||(a[115]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mn",null,"3"),t("mn",null,"4")]),t("mo",null,"="),t("mn",null,"81")])],-1))]),a[129]||(a[129]=Q(" solutions ")),a[130]||(a[130]=t("a",{href:"https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem",target:"_blank",rel:"noreferrer"},"may appear",-1)),a[131]||(a[131]=Q("!"))]),a[168]||(a[168]=e(`
    julia
    result = get_steady_states(harmonic_eq, varied, fixed)
+plot(result, "sqrt(u1^2 + v1^2)")

    ',2)),t("p",null,[a[136]||(a[136]=Q("For the above parameters (where a perturbative treatment would have been reasonable), the principal response at ")),t("mjx-container",w1,[(s(),T("svg",x1,a[132]||(a[132]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[133]||(a[133]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[137]||(a[137]=Q(" looks rather similar, with a much smaller upconverted component appearing at ")),t("mjx-container",y1,[(s(),T("svg",f1,a[134]||(a[134]=[e('',1)]))),a[135]||(a[135]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω")])],-1))]),a[138]||(a[138]=Q(":"))]),a[169]||(a[169]=e(`
    julia
    p1=plot(result, "sqrt(u1^2 + v1^2)", legend=false)
+p2=plot(result, "sqrt(u2^2 + v2^2)")
+plot(p1, p2)

    ',2)),t("p",null,[a[143]||(a[143]=Q("The non-perturbative nature of the ansatz allows us to capture some behaviour which is ")),a[144]||(a[144]=t("em",null,"not",-1)),a[145]||(a[145]=Q(" a mere extension of the usual single-harmonic Duffing response. Suppose we drive a strongly nonlinear resonator at frequency ")),t("mjx-container",V1,[(s(),T("svg",L1,a[139]||(a[139]=[e('',1)]))),a[140]||(a[140]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω"),t("mo",null,"≅"),t("msub",null,[t("mi",null,"ω"),t("mn",null,"0")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mn",null,"3")])],-1))]),a[146]||(a[146]=Q(". Such a drive is far out of resonance, however, the upconverted harmonic ")),t("mjx-container",M1,[(s(),T("svg",v1,a[141]||(a[141]=[e('',1)]))),a[142]||(a[142]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω"),t("mo",null,"="),t("msub",null,[t("mi",null,"ω"),t("mn",null,"0")])])],-1))]),a[147]||(a[147]=Q(" is not and may play an important role! Let us try this out:"))]),a[170]||(a[170]=e(`
    julia
    fixed = (α => 10., ω0 => 3, F => 5, γ=>0.01)   # fixed parameters
+varied = ω => range(0.9, 1.4, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 100 parameter points
+
+Solution branches:   9
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels
    `,2)),t("p",null,[a[152]||(a[152]=Q("Although 9 branches were found in total, only 3 remain physical (real-valued). Let us visualise the amplitudes corresponding to the two harmonics, ")),t("mjx-container",b1,[(s(),T("svg",Z1,a[148]||(a[148]=[e('',1)]))),a[149]||(a[149]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msqrt",null,[t("msubsup",null,[t("mi",null,"U"),t("mn",null,"1"),t("mn",null,"2")]),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"V"),t("mn",null,"1"),t("mn",null,"2")])])])],-1))]),a[153]||(a[153]=Q(" and ")),t("mjx-container",E1,[(s(),T("svg",D1,a[150]||(a[150]=[e('',1)]))),a[151]||(a[151]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msqrt",null,[t("msubsup",null,[t("mi",null,"U"),t("mn",null,"2"),t("mn",null,"2")]),t("mo",null,"+"),t("msubsup",null,[t("mi",null,"V"),t("mn",null,"2"),t("mn",null,"2")])])])],-1))]),a[154]||(a[154]=Q(" :"))]),a[171]||(a[171]=e(`
    julia
    p1 = plot(result, "sqrt(u1^2 + v1^2)", legend=false)
+p2 = plot(result, "sqrt(u2^2 + v2^2)")
+plot(p1, p2)

    ',2)),t("p",null,[a[159]||(a[159]=Q("The contributions of ")),t("mjx-container",C1,[(s(),T("svg",j1,a[155]||(a[155]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),a[156]||(a[156]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"ω")])],-1))]),a[160]||(a[160]=Q(" and ")),t("mjx-container",F1,[(s(),T("svg",B1,a[157]||(a[157]=[e('',1)]))),a[158]||(a[158]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"3"),t("mi",null,"ω")])],-1))]),a[161]||(a[161]=Q(" are now comparable and the system shows some fairly complex behaviour! This demonstrates how an exact solution within an extended Fourier subspace goes beyond a perturbative treatment."))])])}const R1=l(d,[["render",A1]]);export{N1 as __pageData,R1 as default}; diff --git a/v0.10.7/assets/tutorials_time_dependent.md.DKDMpmEj.js b/v0.10.7/assets/tutorials_time_dependent.md.DKDMpmEj.js new file mode 100644 index 00000000..25be6002 --- /dev/null +++ b/v0.10.7/assets/tutorials_time_dependent.md.DKDMpmEj.js @@ -0,0 +1,36 @@ +import{_ as l,c as e,a4 as t,j as s,a,o as n}from"./chunks/framework.D3a-l5sU.js";const h="/HarmonicBalance.jl/v0.10.7/assets/nmezmfs.CdNyJeEn.png",p="/HarmonicBalance.jl/v0.10.7/assets/frnvjsq.C1saRSuo.png",o="/HarmonicBalance.jl/v0.10.7/assets/cqedvtw.dPeTlm0F.png",r="/HarmonicBalance.jl/v0.10.7/assets/qbfgfrf.CbjDi_ic.png",q=JSON.parse('{"title":"Time-dependent simulations","description":"","frontmatter":{},"headers":[],"relativePath":"tutorials/time_dependent.md","filePath":"tutorials/time_dependent.md"}'),d={name:"tutorials/time_dependent.md"},k={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},Q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.575ex"},xmlns:"http://www.w3.org/2000/svg",width:"15.838ex",height:"4.878ex",role:"img",focusable:"false",viewBox:"0 -1460 7000.6 2156","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},T={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.799ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2121 1000","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.799ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2121 1000","aria-hidden":"true"},E={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.515ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2437.6 1000","aria-hidden":"true"},u={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},x={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.799ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2121 1000","aria-hidden":"true"},w={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},v={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.316ex",height:"1.692ex",role:"img",focusable:"false",viewBox:"0 -666 3233.6 748","aria-hidden":"true"},f={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},F={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.316ex",height:"1.692ex",role:"img",focusable:"false",viewBox:"0 -666 3233.6 748","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},H={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},C={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},D={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.447ex",height:"1.692ex",role:"img",focusable:"false",viewBox:"0 -666 3733.6 748","aria-hidden":"true"};function L(M,i,V,B,A,j){return n(),e("div",null,[i[32]||(i[32]=t('

    Time-dependent simulations ​

    Most of HarmonicBalance.jl is focused on finding and analysing the steady states. Such states contain no information about transient behaviour, which is crucial to answer the following.

    • Given an initial condition, which steady state does the system evolve into?

    • How does the system behave if its parameters are varied in time?

    It is straightforward to evolve the full equation of motion using an ODE solver. However, tracking oscillatory behaviour is computationally expensive.

    In the background, we showed that nonlinear driven systems may be reduced to harmonic equations

    ',5)),s("mjx-container",k,[(n(),e("svg",Q,i[0]||(i[0]=[t('',1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mfrac",null,[s("mrow",null,[s("mi",null,"d"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mi",null,"d"),s("mi",null,"T")])]),s("mo",null,"="),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"F")]),s("mo",{stretchy:"false"},"¯")])]),s("mo",{stretchy:"false"},"("),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},")"),s("mstyle",{scriptlevel:"0"},[s("mspace",{width:"0.167em"})]),s("mo",null,",")])],-1))]),s("p",null,[i[6]||(i[6]=a("As long as the chosen harmonics constituting ")),s("mjx-container",m,[(n(),e("svg",T,i[2]||(i[2]=[t('',1)]))),i[3]||(i[3]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")")])],-1))]),i[7]||(i[7]=a(" capture the system's behaviour, we may numerically evolve the new effective differential equations instead of the full problem. Since the components of ")),s("mjx-container",g,[(n(),e("svg",c,i[4]||(i[4]=[t('',1)]))),i[5]||(i[5]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")")])],-1))]),i[8]||(i[8]=a(" only vary very slowly (and are constant in a steady state), this is usually ")),i[9]||(i[9]=s("em",null,"vastly",-1)),i[10]||(i[10]=a(" more efficient than evolving the full problem."))]),i[33]||(i[33]=t(`

    Here we primarily demonstrate on the parametrically driven oscillator.

    We start by defining our system.

    julia
    using HarmonicBalance
+@variables ω0 γ λ F θ η α ω t x(t)
+
+eq =  d(d(x,t),t) + γ*d(x,t) + ω0^2*(1 - λ*cos(2*ω*t))*x + α*x^3 + η*d(x,t)*x^2 ~ F*cos(ω*t + θ)
+
+diff_eq = DifferentialEquation(eq, x)
+add_harmonic!(diff_eq, x, ω); # single-frequency ansatz
+
+harmonic_eq = get_harmonic_equations(diff_eq);
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, λ, ω0, η, θ, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*Differential(T)(u1(T))*η + (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*η + (3//4)*u1(T)*(v1(T)^2)*α - (1//2)*u1(T)*λ*(ω0^2) + (1//4)*(v1(T)^2)*Differential(T)(u1(T))*η + (1//4)*(u1(T)^2)*v1(T)*η*ω + (1//4)*(v1(T)^3)*η*ω ~ F*cos(θ)
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*η + (3//4)*(u1(T)^2)*v1(T)*α + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*η + (3//4)*Differential(T)(v1(T))*(v1(T)^2)*η + (3//4)*(v1(T)^3)*α + (1//2)*v1(T)*λ*(ω0^2) - (1//4)*(u1(T)^3)*η*ω - (1//4)*u1(T)*(v1(T)^2)*η*ω ~ -F*sin(θ)

    The object harmonic_eq encodes the new effective differential equations.

    We now wish to parse this input into OrdinaryDiffEq.jl and use its powerful ODE solvers. The desired object here is OrdinaryDiffEq.ODEProblem, which is then fed into OrdinaryDiffEq.solve.

    Evolving from an initial condition ​

    `,7)),s("p",null,[i[15]||(i[15]=a("Given ")),s("mjx-container",E,[(n(),e("svg",y,i[11]||(i[11]=[t('',1)]))),i[12]||(i[12]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"T"),s("mn",null,"0")]),s("mo",{stretchy:"false"},")")])],-1))]),i[16]||(i[16]=a(", what is ")),s("mjx-container",u,[(n(),e("svg",x,i[13]||(i[13]=[t('',1)]))),i[14]||(i[14]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")")])],-1))]),i[17]||(i[17]=a(" at future times?"))]),i[34]||(i[34]=t(`

    For constant parameters, a HarmonicEquation object can be fed into the constructor of ODEProblem. The syntax is similar to DifferentialEquations.jl :

    julia
    using OrdinaryDiffEq
+x0 = [0.; 0.] # initial condition
+fixed = (ω0 => 1.0, γ => 1e-2, λ => 5e-2, F => 1e-3,  α => 1.0, η => 0.3, θ => 0, ω => 1.0) # parameter values
+
+ode_problem = ODEProblem(harmonic_eq, fixed, x0 = x0, timespan = (0,1000))
    ODEProblem with uType Vector{Float64} and tType Int64. In-place: true
+timespan: (0, 1000)
+u0: 2-element Vector{Float64}:
+ 0.0
+ 0.0

    OrdinaryDiffEq.jl takes it from here - we only need to use solve.

    julia
    time_evo = solve(ode_problem, saveat=1.0);
+plot(time_evo, ["u1", "v1"], harmonic_eq)

    Running the above code with x0 = [0.2, 0.2] gives the plots

    julia
    x0 = [0.2; 0.2] # initial condition
+ode_problem = remake(ode_problem, u0 = x0)
+time_evo = solve(ode_problem, saveat=1.0);
+plot(time_evo, ["u1", "v1"], harmonic_eq)

    Let us compare this to the steady state diagram.

    julia
    varied = ω => range(0.9, 1.1, 100)
+result = get_steady_states(harmonic_eq, varied, fixed)
+plot(result, "sqrt(u1^2 + v1^2)")

    Clearly when evolving from x0 = [0.,0.], the system ends up in the low-amplitude branch 2. With x0 = [0.2, 0.2], the system ends up in branch 3.

    Adiabatic parameter sweeps ​

    Experimentally, the primary means of exploring the steady state landscape is an adiabatic sweep one or more of the system parameters. This takes the system along a solution branch. If this branch disappears or becomes unstable, a jump occurs.

    The object ParameterSweep specifies a sweep, which is then used as an optional sweep keyword in the ODEProblem constructor.

    julia
    sweep = ParameterSweep(ω => (0.9,1.1), (0, 2e4))
    ParameterSweep(Dict{Num, Function}(ω => TimeEvolution.var"#f#1"{Tuple{Float64, Float64}, Float64, Int64}((0.9, 1.1), 20000.0, 0)))
    ',18)),s("p",null,[i[24]||(i[24]=a("The sweep linearly interpolates between ")),s("mjx-container",w,[(n(),e("svg",v,i[18]||(i[18]=[t('',1)]))),i[19]||(i[19]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω"),s("mo",null,"="),s("mn",null,"0.9")])],-1))]),i[25]||(i[25]=a(" at time 0 and ")),s("mjx-container",f,[(n(),e("svg",F,i[20]||(i[20]=[t('',1)]))),i[21]||(i[21]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω"),s("mo",null,"="),s("mn",null,"1.1")])],-1))]),i[26]||(i[26]=a(" at time 2e4. For earlier/later times, ")),s("mjx-container",b,[(n(),e("svg",H,i[22]||(i[22]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),i[23]||(i[23]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω")])],-1))]),i[27]||(i[27]=a(" is constant."))]),i[35]||(i[35]=t(`

    Let us now define a new ODEProblem which incorporates sweep and again use solve:

    julia
    ode_problem = ODEProblem(harmonic_eq, fixed, sweep=sweep, x0=[0.1;0.0], timespan=(0, 2e4))
+time_evo = solve(ode_problem, saveat=100)
+plot(time_evo, "sqrt(u1^2 + v1^2)", harmonic_eq)

    ',3)),s("p",null,[i[30]||(i[30]=a("We see the system first evolves from the initial condition towards the low-amplitude steady state. The amplitude increases as the sweep proceeds, with a jump occurring around ")),s("mjx-container",C,[(n(),e("svg",D,i[28]||(i[28]=[t('',1)]))),i[29]||(i[29]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω"),s("mo",null,"="),s("mn",null,"1.08")])],-1))]),i[31]||(i[31]=a(" (i.e., time 18000)."))])])}const O=l(d,[["render",L]]);export{q as __pageData,O as default}; diff --git a/v0.10.7/assets/tutorials_time_dependent.md.DKDMpmEj.lean.js b/v0.10.7/assets/tutorials_time_dependent.md.DKDMpmEj.lean.js new file mode 100644 index 00000000..25be6002 --- /dev/null +++ b/v0.10.7/assets/tutorials_time_dependent.md.DKDMpmEj.lean.js @@ -0,0 +1,36 @@ +import{_ as l,c as e,a4 as t,j as s,a,o as n}from"./chunks/framework.D3a-l5sU.js";const h="/HarmonicBalance.jl/v0.10.7/assets/nmezmfs.CdNyJeEn.png",p="/HarmonicBalance.jl/v0.10.7/assets/frnvjsq.C1saRSuo.png",o="/HarmonicBalance.jl/v0.10.7/assets/cqedvtw.dPeTlm0F.png",r="/HarmonicBalance.jl/v0.10.7/assets/qbfgfrf.CbjDi_ic.png",q=JSON.parse('{"title":"Time-dependent simulations","description":"","frontmatter":{},"headers":[],"relativePath":"tutorials/time_dependent.md","filePath":"tutorials/time_dependent.md"}'),d={name:"tutorials/time_dependent.md"},k={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},Q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.575ex"},xmlns:"http://www.w3.org/2000/svg",width:"15.838ex",height:"4.878ex",role:"img",focusable:"false",viewBox:"0 -1460 7000.6 2156","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},T={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.799ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2121 1000","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.799ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2121 1000","aria-hidden":"true"},E={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.515ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2437.6 1000","aria-hidden":"true"},u={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},x={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.799ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2121 1000","aria-hidden":"true"},w={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},v={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.316ex",height:"1.692ex",role:"img",focusable:"false",viewBox:"0 -666 3233.6 748","aria-hidden":"true"},f={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},F={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.316ex",height:"1.692ex",role:"img",focusable:"false",viewBox:"0 -666 3233.6 748","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},H={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},C={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},D={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.447ex",height:"1.692ex",role:"img",focusable:"false",viewBox:"0 -666 3733.6 748","aria-hidden":"true"};function L(M,i,V,B,A,j){return n(),e("div",null,[i[32]||(i[32]=t('

    Time-dependent simulations ​

    Most of HarmonicBalance.jl is focused on finding and analysing the steady states. Such states contain no information about transient behaviour, which is crucial to answer the following.

    • Given an initial condition, which steady state does the system evolve into?

    • How does the system behave if its parameters are varied in time?

    It is straightforward to evolve the full equation of motion using an ODE solver. However, tracking oscillatory behaviour is computationally expensive.

    In the background, we showed that nonlinear driven systems may be reduced to harmonic equations

    ',5)),s("mjx-container",k,[(n(),e("svg",Q,i[0]||(i[0]=[t('',1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mfrac",null,[s("mrow",null,[s("mi",null,"d"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")")]),s("mrow",null,[s("mi",null,"d"),s("mi",null,"T")])]),s("mo",null,"="),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"F")]),s("mo",{stretchy:"false"},"¯")])]),s("mo",{stretchy:"false"},"("),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},")"),s("mstyle",{scriptlevel:"0"},[s("mspace",{width:"0.167em"})]),s("mo",null,",")])],-1))]),s("p",null,[i[6]||(i[6]=a("As long as the chosen harmonics constituting ")),s("mjx-container",m,[(n(),e("svg",T,i[2]||(i[2]=[t('',1)]))),i[3]||(i[3]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")")])],-1))]),i[7]||(i[7]=a(" capture the system's behaviour, we may numerically evolve the new effective differential equations instead of the full problem. Since the components of ")),s("mjx-container",g,[(n(),e("svg",c,i[4]||(i[4]=[t('',1)]))),i[5]||(i[5]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")")])],-1))]),i[8]||(i[8]=a(" only vary very slowly (and are constant in a steady state), this is usually ")),i[9]||(i[9]=s("em",null,"vastly",-1)),i[10]||(i[10]=a(" more efficient than evolving the full problem."))]),i[33]||(i[33]=t(`

    Here we primarily demonstrate on the parametrically driven oscillator.

    We start by defining our system.

    julia
    using HarmonicBalance
+@variables ω0 γ λ F θ η α ω t x(t)
+
+eq =  d(d(x,t),t) + γ*d(x,t) + ω0^2*(1 - λ*cos(2*ω*t))*x + α*x^3 + η*d(x,t)*x^2 ~ F*cos(ω*t + θ)
+
+diff_eq = DifferentialEquation(eq, x)
+add_harmonic!(diff_eq, x, ω); # single-frequency ansatz
+
+harmonic_eq = get_harmonic_equations(diff_eq);
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, λ, ω0, η, θ, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*Differential(T)(u1(T))*η + (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*η + (3//4)*u1(T)*(v1(T)^2)*α - (1//2)*u1(T)*λ*(ω0^2) + (1//4)*(v1(T)^2)*Differential(T)(u1(T))*η + (1//4)*(u1(T)^2)*v1(T)*η*ω + (1//4)*(v1(T)^3)*η*ω ~ F*cos(θ)
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*η + (3//4)*(u1(T)^2)*v1(T)*α + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*η + (3//4)*Differential(T)(v1(T))*(v1(T)^2)*η + (3//4)*(v1(T)^3)*α + (1//2)*v1(T)*λ*(ω0^2) - (1//4)*(u1(T)^3)*η*ω - (1//4)*u1(T)*(v1(T)^2)*η*ω ~ -F*sin(θ)

    The object harmonic_eq encodes the new effective differential equations.

    We now wish to parse this input into OrdinaryDiffEq.jl and use its powerful ODE solvers. The desired object here is OrdinaryDiffEq.ODEProblem, which is then fed into OrdinaryDiffEq.solve.

    Evolving from an initial condition ​

    `,7)),s("p",null,[i[15]||(i[15]=a("Given ")),s("mjx-container",E,[(n(),e("svg",y,i[11]||(i[11]=[t('',1)]))),i[12]||(i[12]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},"("),s("msub",null,[s("mi",null,"T"),s("mn",null,"0")]),s("mo",{stretchy:"false"},")")])],-1))]),i[16]||(i[16]=a(", what is ")),s("mjx-container",u,[(n(),e("svg",x,i[13]||(i[13]=[t('',1)]))),i[14]||(i[14]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"bold"},"u")]),s("mo",{stretchy:"false"},"("),s("mi",null,"T"),s("mo",{stretchy:"false"},")")])],-1))]),i[17]||(i[17]=a(" at future times?"))]),i[34]||(i[34]=t(`

    For constant parameters, a HarmonicEquation object can be fed into the constructor of ODEProblem. The syntax is similar to DifferentialEquations.jl :

    julia
    using OrdinaryDiffEq
+x0 = [0.; 0.] # initial condition
+fixed = (ω0 => 1.0, γ => 1e-2, λ => 5e-2, F => 1e-3,  α => 1.0, η => 0.3, θ => 0, ω => 1.0) # parameter values
+
+ode_problem = ODEProblem(harmonic_eq, fixed, x0 = x0, timespan = (0,1000))
    ODEProblem with uType Vector{Float64} and tType Int64. In-place: true
+timespan: (0, 1000)
+u0: 2-element Vector{Float64}:
+ 0.0
+ 0.0

    OrdinaryDiffEq.jl takes it from here - we only need to use solve.

    julia
    time_evo = solve(ode_problem, saveat=1.0);
+plot(time_evo, ["u1", "v1"], harmonic_eq)

    Running the above code with x0 = [0.2, 0.2] gives the plots

    julia
    x0 = [0.2; 0.2] # initial condition
+ode_problem = remake(ode_problem, u0 = x0)
+time_evo = solve(ode_problem, saveat=1.0);
+plot(time_evo, ["u1", "v1"], harmonic_eq)

    Let us compare this to the steady state diagram.

    julia
    varied = ω => range(0.9, 1.1, 100)
+result = get_steady_states(harmonic_eq, varied, fixed)
+plot(result, "sqrt(u1^2 + v1^2)")

    Clearly when evolving from x0 = [0.,0.], the system ends up in the low-amplitude branch 2. With x0 = [0.2, 0.2], the system ends up in branch 3.

    Adiabatic parameter sweeps ​

    Experimentally, the primary means of exploring the steady state landscape is an adiabatic sweep one or more of the system parameters. This takes the system along a solution branch. If this branch disappears or becomes unstable, a jump occurs.

    The object ParameterSweep specifies a sweep, which is then used as an optional sweep keyword in the ODEProblem constructor.

    julia
    sweep = ParameterSweep(ω => (0.9,1.1), (0, 2e4))
    ParameterSweep(Dict{Num, Function}(ω => TimeEvolution.var"#f#1"{Tuple{Float64, Float64}, Float64, Int64}((0.9, 1.1), 20000.0, 0)))
    ',18)),s("p",null,[i[24]||(i[24]=a("The sweep linearly interpolates between ")),s("mjx-container",w,[(n(),e("svg",v,i[18]||(i[18]=[t('',1)]))),i[19]||(i[19]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω"),s("mo",null,"="),s("mn",null,"0.9")])],-1))]),i[25]||(i[25]=a(" at time 0 and ")),s("mjx-container",f,[(n(),e("svg",F,i[20]||(i[20]=[t('',1)]))),i[21]||(i[21]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω"),s("mo",null,"="),s("mn",null,"1.1")])],-1))]),i[26]||(i[26]=a(" at time 2e4. For earlier/later times, ")),s("mjx-container",b,[(n(),e("svg",H,i[22]||(i[22]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D714",d:"M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z",style:{"stroke-width":"3"}})])])],-1)]))),i[23]||(i[23]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω")])],-1))]),i[27]||(i[27]=a(" is constant."))]),i[35]||(i[35]=t(`

    Let us now define a new ODEProblem which incorporates sweep and again use solve:

    julia
    ode_problem = ODEProblem(harmonic_eq, fixed, sweep=sweep, x0=[0.1;0.0], timespan=(0, 2e4))
+time_evo = solve(ode_problem, saveat=100)
+plot(time_evo, "sqrt(u1^2 + v1^2)", harmonic_eq)

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The amplitude increases as the sweep proceeds, with a jump occurring around ")),s("mjx-container",C,[(n(),e("svg",D,i[28]||(i[28]=[t('',1)]))),i[29]||(i[29]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω"),s("mo",null,"="),s("mn",null,"1.08")])],-1))]),i[31]||(i[31]=a(" (i.e., time 18000)."))])])}const O=l(d,[["render",L]]);export{q as __pageData,O as default}; diff --git a/v0.10.7/assets/ucpiacu.UTcoxLl5.png b/v0.10.7/assets/ucpiacu.UTcoxLl5.png new file mode 100644 index 0000000000000000000000000000000000000000..74a9001a0b24d60073629028056e09304090d48e GIT binary patch literal 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HarmonicBalance.jl + + + + + + + + + + + + + + + +

    HarmonicBalance.jl

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    The method of harmonic balance ​

    Frequency conversion in oscillating nonlinear systems ​

    HarmonicBalance.jl focuses on harmonically-driven nonlinear systems, i.e., dynamical systems governed by equations of motion where all explicitly time-dependent terms are harmonic. Let us take a general nonlinear system of N second-order ODEs with real variables xi(t), i=1,2,⋯,N and time t as the independent variable,

    x¨(t)+F(x(t),t)=0.

    The vector x(t)=(x1(t),...,xN(t))T fully describes the state of the system. Physically, x(t) encompasses the amplitudes of either point-like or collective oscillators (e.g., mechanical resonators, voltage oscillations in RLC circuits, an oscillating electrical dipole moment, or standing modes of an optical cavity).

    As the simplest example, let us first solve the harmonic oscillator in frequency space. The equation of motion is

    x¨(t)+γx˙(t)+ω02x(t)=Fcos⁡(ωdt)

    where γ is the damping coefficient and ω0 the natural frequency. Fourier-transforming both sides of this equation gives

    (ω02−ω2+iωγ)x~(ω)=F2[δ(ω+ωd)+δ(ω−ωd)].

    Evidently, x~(ω) is only nonvanishing for ω=±ωd. The system thus responds at the driving frequency only - the behaviour can be captured by a single harmonic. This illustrates the general point that linear systems are exactly solvable by transforming to Fourier space, where the equations are diagonal.

    The situation becomes more complex if nonlinear terms are present, as these cause frequency conversion. Suppose we add a quadratic nonlinearity βx2(t) to the equations of motion; an attempt to Fourier-transform gives

    FT[x2](ω)=∫x2(t)e−iωtdt=∫−∞+∞x~(ω′)x~(ω″)δ(ω″+ω′−ω)dω′dω″,

    which couples all harmonics ω,ω′,ω″ such that ω+ω′+ω″=0. To lowest order, this means the induced motion at the drive frequency generates a higher harmonic, ωd→2ωd. To higher orders however, the frequency conversion propagates through the spectrum, coupling an infinite number of harmonics. The system is not solvable in Fourier space anymore!

    Harmonic ansatz & harmonic equations ​

    Even though we need an infinity of Fourier components to describe our system exactly, some components are more important than others. The strategy of harmonic balance is to describe the motion of any variable xi(t) in a truncated Fourier space

    xi(t)=∑j=1Miui,j(T)cos⁡(ωi,jt)+vi,j(T)sin⁡(ωi,jt).

    Within this space, the system is described by a finite-dimensional vector

    u(T)=(u1,1(T),v1,1(T),…uN,MN(T),vN,MN(T))

    Under the assumption that u(T) evolves at much slower timescales than the oscillatory terms ωi,jt, we may neglect all of its higher order time derivatives. Notice that once ansatz \eqref{eq:harmansatz} is used in Eq. \eqref{eq:ode}, all terms become oscillatory - each prefactor of cos⁡(ωi,jt) and sin⁡(ωi,jt) thus generates a separate equation. Collecting these, we obtain a 1st order nonlinear ODEs,

    du(T)dT=F¯(u),

    which we call the harmonic equations. The main purpose of HarmonicBalance.jl is to obtain and solve them. We are primarily interested in steady states u0 defined by F¯(u0)=0.

    The process of obtaining the harmonic equations is best shown on an example.

    Example: the Duffing oscillator ​

    Here, we derive the harmonic equations for a single Duffing resonator, governed by the equation

    x¨(t)+ω02x(t)+αx3(t)=Fcos⁡(ωdt+θ).

    As explained in above, for a periodic driving at frequency ωd and a weak nonlinearity α, we expect the response at frequency ωd to dominate, followed by a response at 3ωd due to frequency conversion.

    Single-frequency ansatz ​

    We first attempt to describe the steady states of Eq. \eqref{eq:duffing} using only one harmonic, ωd. The starting point is the harmonic ansatz for x

    x(t)=u(T)cos⁡(ωdt)+v(T)sin⁡(ωdt),

    with the harmonic variables u and v. The slow time T is, for now, equivalent to t. Substituting this ansatz into mechanical equations of motion results in

    [u¨+2ωdv˙+u(ω02−ωd2)+3α(u3+uv2)4+Fcos⁡θ]cos⁡(ωdt)+[v¨−2ωdu˙+v(ω02−ωd2)+3α(v3+u2v)4−Fsin⁡θ]sin⁡(ωdt)+α(u3−3uv2)4cos⁡(3ωdt)+α(3u2v−v3)4sin⁡(3ωdt)=0.

    We see that the x3 term has generated terms that oscillate at 3ωd, describing the process of frequency upconversion. We now Fourier-transform both sides of Eq. \eqref{eq:ansatz1} with respect to ωd to obtain the harmonic equations. This process is equivalent to extracting the respective coefficients of cos⁡(ωdt) and sin⁡(ωdt). Here the distinction between t and T becomes important: since the evolution of u(T) and v(T) is assumed to be slow, they are treated as constant for the purpose of the Fourier transformation. Since we are interested in steady states, we drop the higher-order derivatives and rearrange the resulting equation to

    ddT(uv)=18ωd(4v(ω02−ωd2)+3α(v3+u2v)−4Fsin⁡θ4u(ωd2−ω02)−3α(u3+uv2)−4Fcos⁡θ).

    Steady states can now be found by setting the l.h.s. to zero, i.e., assuming u(T) and v(T) constant and neglecting any transient behaviour. This results in a set of 2 nonlinear polynomial equations of order 3, for which the maximum number of solutions set by Bézout's theorem is 32=9. Depending on the parameters, the number of real solutions is known to be between 1 and 3.

    Sidenote: perturbative approach ​

    The steady states describe a response that may be recast as x0(t)=X0cos⁡(ωdt+ϕ), where X0=u2+v2 and ϕ=−atan(v/u). Frequency conversion from ωd to 3ωd can be found by setting x(t)≡x0(t)+δx(t) with |δx(t)|≪|x0(t)| and expanding Eq. \eqref{eq:duffing} to first-order in δx(t). The resulting equation

    δx¨(t)+[ω02+3αX024]δx(t)=−αX034cos⁡(3ωdt+3ϕ),

    describes a simple harmonic oscillator, which is exactly soluble. Correspondingly, a response of δx(t) at frequency 3ωd is observed. Since this response is obtained 'on top of' each steady state of the equations of motion, no previously-unknown solutions are generated in the process.

    Two-frequency ansatz ​

    An approach in the spirit of harmonic balance is to use both harmonics ωd and 3ωd on the same footing, i.e., to insert the ansatz

    x(t)=u1(T)cos⁡(ωdt)+v1(T)sin⁡(ωdt)+u2(T)cos⁡(3ωdt)+v2(T)sin⁡(3ωdt),

    with u1,u2,v1,v2 being the harmonic variables. As before we substitute the ansatz into Eq. \eqref{eq:duffing}, drop second derivatives with respect to T and Fourier-transform both sides. Now, the respective coefficients correspond to cos⁡(ωdt), sin⁡(ωdt), cos⁡(3ωdt) and sin⁡(3ωdt). Rearranging, we obtain

    du1dT=12ωd[(ω02−ωd2)v1+3α4(v13+u12v1+u12v2−v12v2+2u22v1+2v22v1−2u1u2v1)+Fsin⁡θ],dv1dT=12ωd[(ωd2−ω02)u1−3α4(u13+u12u2+v12u1−v12u2+2u22u1+2v22u1+2u1v1v2)−Fcos⁡θ],du2dT=16ωd[(ω02−9ωd2)v2+α4(−v13+3v23+3u12v1+6u12v2+3u22v2+6v12v2)],dv2dT=16ωd[(9ωd2−ω02)u2−α4(u13+3u23+6u12u2−3v12u1+3v22u2+6v12u2)].

    In contrast to the single-frequency ansatz, we now have 4 equations of order 3, allowing up to 34=81 solutions (the number of unique real ones is again generally far smaller). The larger number of solutions is explained by higher harmonics which cannot be captured perturbatively by the single-frequency ansatz. In particular, those where the 3ωd component is significant. Such solutions appear, e.g., for ωd≈ω0/3 where the generated 3ωd harmonic is close to the natural resonant frequency. See the examples for numerical results.

    + + + + \ No newline at end of file diff --git a/v0.10.7/background/limit_cycles.html b/v0.10.7/background/limit_cycles.html new file mode 100644 index 00000000..3c7337c7 --- /dev/null +++ b/v0.10.7/background/limit_cycles.html @@ -0,0 +1,26 @@ + + + + + + Limit cycles | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

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    Limit cycles ​

    We explain how HarmonicBalance.jl uses a new technique to find limit cycles in systems of nonlinear ODEs. For a more in depth overwiew see Chapter 6 in Jan Košata's PhD theses or del_Pino_2024.

    Limit cycles from a Hopf bifurcation ​

    The end product of the harmonic balance technique are what we call the harmonic equations, i.e., first-order ODEs for the harmonic variables U(T):

    dU(T)dT=G―(U)

    These Odes have no explicit time-dependence - they are autonomous. We have mostly been searching for steady states, which likewise show no time dependence. However, time-dependent solutions to autonomous ODEs can also exist. One mechanism for their creation is a Hopf bifurcation - a critical point where a stable solution transitions into an unstable one. For a stable solution, the associated eigenvalues λ of the linearisation all satisfy Re(λ)<0. When a Hopf bifurcation takes place, one complex-conjugate pair of eigenvalues crosses the real axis such that Re(λ)>0. The state is then, strictly speaking, unstable. However, instead of evolving into another steady state, the system may assume a periodic orbit in phase space, giving a solution of the form

    U(T)=U0+Ulccos⁡(ωlcT+ϕ)

    which is an example of a limit cycle. We denote the originating steady state as Hopf-unstable.

    We can continue to use harmonic balance as the solution still describes a harmonic response Allwright (1977). If we translate back to the the lab frame [variable x(t)], clearly, each frequency ωj constituting our harmonic ansatz [U(T)], we obtain frequencies ωj as well as ωj±ωlc  in the lab frame. Furthermore, as multiple harmonics now co-exist in the system, frequency conversion may take place, spawning further pairs ωj±kωlc  with integer k. Therefore, to construct a harmonic ansatz capturing limit cycles, we simply add an integer number K of such pairs to our existing set of M harmonics,

    {ω1,…,ωM}→{ω1,ω1±ωlc,ω1±2ωlc,…,ωM±Kωlc}

    Ansatz ​

    Original ansatz ​

    Having seen how limit cycles are formed, we now proceed to tackle a key problem: how to find their frequency ωlc. We again demonstrate by considering a single variable x(t). We may try the simplest ansatz for a system driven at frequency ω,

    x(t)=u1(T)cos⁡(ωt)+v1(T)sin⁡(ωt)

    In this formulation, limit cycles may be obtained by solving the resulting harmonic equations with a Runge-Kutta type solver to obtain the time evolution of u1(T) and v1(T). See the limit cycle tutorial for an example.

    Extended ansatz ​

    Including newly-emergent pairs of harmonics is in principle straightforward. Suppose a limit cycle has formed in our system with a frequency ωlc, prompting the ansatz

    x(t)=u1cos⁡(ωt)+v1sin⁡(ωt)+u2cos⁡[(ω+ωlc)t]+v2sin⁡[(ω+ωlc)t]+u3cos⁡[(ω−ωlc)t]+v3sin⁡[(ω−ωlc)t]+…

    where each of the ω±kωlc  pairs contributes 4 harmonic variables. The limit cycle frequency ωlc is also a variable in this formulation, but does not contribute a harmonic equation, since dωlc/dT=0 by construction. We thus arrive at a total of 2+4K harmonic equations in 2+4K+1 variables. To obtain steady states, we must thus solve an underdetermined system, which has an infinite number of solutions. Given that we expect the limit cycles to possess U(1) gauge freedom, this is a sensible observation. We may still use iterative numerical procedures such as the Newton method to find solutions one by one, but homotopy continuation is not applicable. In this formulation, steady staes states are characterised by zero entries for u2,v2,…u2K+1,v2K+1. The variable ωlc  is redundant and may take any value - the states therefore also appear infinitely degenerate, which, however, has no physical grounds. Oppositely, solutions may appear for which some of the limit cycle variables u2,v2,…u2K+1,v2K+1 are nonzero, but ωlc =0. These violate our assumption of distinct harmonic variables corresponding to distinct frequencies and are therefore discarded.

    Gauge fixing ​

    We now constrain the system to remove the U(1) gauge freedom. This is best done by explicitly writing out the free phase. Recall that our solution must be symmetric under a time translation symmetry, that is, taking t→t+2π/ω. Applying this n times transforms x(t) into

    x(t)=u1cos⁡(ωt)+v1sin⁡(ωt)+u2cos⁡[(ω+ωlc)t+ϕ]+v2sin⁡[(ω+ωlc)t+ϕ]+u3cos⁡[(ω−ωlc)t−ϕ]+v3sin⁡[(ω−ωlc)t−ϕ]+…

    where we defined ϕ=2πnωlc /ω. Since ϕ is free, we can fix it to, for example,

    ϕ=−arctan⁡u2/v2

    which turns into

    x(t)=u1cos⁡(ωt)+v1sin⁡(ωt)+(v2cos⁡ϕ−u2sin⁡ϕ)sin⁡[(ω+ωlc)t]+(u3cos⁡ϕ−v3sin⁡ϕ)cos⁡[(ω−ωlc)t]+(v3cos⁡ϕ+u3sin⁡ϕ)[(ω−ωlc)t]+…

    We see that fixing the free phase has effectively removed one of the variables, since cos⁡[(ω+ωlc )t] does not appear any more. Discarding u2, we can therefore use 2+4K variables as our harmonic ansatz, i.e.,

    U=(u1v1v2⋮v2K+1ωlc)

    to remove the infinite degeneracy. Note that ϕ is only defined modulo π, but its effect on the harmonic variables is not. Choosing ϕ=−arctan⁡u2/v2+π would invert the signs of v2,u3,v3. As a result, each solution is doubly degenerate. Combined with the sign ambiguity of ωlc , we conclude that under the new ansatz, a limit cycle solution appears as a fourfold-degenerate steady state.

    The harmonic equations can now be solved using homotopy continuation to obtain all steady states. Compared to the single-harmonic ansatz however, we have significantly enlarged the polynomial system to be solved. As the number of solutions scales exponentially (Bézout bound), we expect vast numbers of solutions even for fairly small systems.

    + + + + \ No newline at end of file diff --git a/v0.10.7/background/stability_response.html b/v0.10.7/background/stability_response.html new file mode 100644 index 00000000..3279089b --- /dev/null +++ b/v0.10.7/background/stability_response.html @@ -0,0 +1,26 @@ + + + + + + Stability and linear response | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

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    Stability and linear response ​

    The core of the harmonic balance method is expressing the system's behaviour in terms of Fourier components or harmonics. For an N-coordinate system, we choose a set of Mi harmonics to describe each coordinate xi :

    xi(t)=∑j=1Miui,j(T)cos⁡(ωi,jt)+vi,j(T)sin⁡(ωi,jt),

    This means the system is now described using a discrete set of variables ui,j and vi,j. Constructing the vector

    u(T)=(u1,1(T),v1,1(T),…uN,MN(T),vN,MN(T)),

    we may obtain the harmonic equations (see an example of this procedure)

    du(T)dT=F¯(u)

    where F¯(u) is a nonlinear function. A steady state u0 is defined by F¯(u0)=0.

    Stability ​

    Let us assume that we found a steady state u0. When the system is in this state, it responds to small perturbations either by returning to u0 over some characteristic timescale (stable state) or by evolving away from u0 (unstable state). To analyze the stability of u0, we linearize the equations of motion around u0 for a small perturbation δu=u−u0 to obtain

    ddT[δu(T)]=J(u0)δu(T),

    where J(u0)=∇uF¯|u=u0 is the Jacobian matrix of the system evaluated at u=u0.

    The linearised system is exactly solvable for δu(T) given an initial condition δu(T0). The solution can be expanded in terms of the complex eigenvalues λr and eigenvectors vr of J(u0), namely

    δu(T)=∑rcrvreλrT.

    The dynamical behaviour near the steady states is thus governed by eλrT: if Re(λr)<0 for all λr, the state u0 is stable. Conversely, if Re(λr)>0 for at least one λr, the state is unstable - perturbations such as noise or a small applied drive will force the system away from u0.

    Linear response ​

    The response of a stable steady state to an additional oscillatory force, caused by weak probes or noise, is often of interest. It can be calculated by solving for the perturbation δu(T) in the presence of an additional drive term.

    ddT[δu(T)]=J(u0)δu(T)+ξeiΩT,

    Suppose we have found an eigenvector of J(u0) such that J(u)v=λv. To solve the linearised equations of motion, we insert δu(T)=A(Ω)veiΩT. Projecting each side onto v gives

    A(Ω)(iΩ−λ)=ξ⋅v⟹A(Ω)=ξ⋅v−Re[λ]+i(Ω−Im[λ])

    We see that each eigenvalue λ results in a linear response that is a Lorentzian centered at Ω=Im[λ]. Effectively, the linear response matches that of a harmonic oscillator with resonance frequency Im[λ] and damping Re[λ].

    Knowing the response of the harmonic variables u(T), what is the corresponding behaviour of the "natural" variables xi(t)? To find this out, we insert the perturbation back into the harmonic ansatz. Since we require real variables, let us use δu(T)=A(Ω)(veiΩT+v∗e−iΩT). Plugging this into

    δxi(t)=∑j=1Miδui,j(t)cos⁡(ωi,jt)+δvi,j(t)sin⁡(ωi,jt)

    and multiplying out the sines and cosines gives

    δxi(t)=∑j=1Mi{(Re[δui,j]−Im[δvi,j])cos⁡[(ωi,j−Ω)t]+(Im[δui,j]+Re[δvi,j])sin⁡[(ωi,j−Ω)t]+(Re[δui,j]+Im[δvi,j])cos⁡[(ωi,j+Ω)t]+(−Im[δui,j]+Re[δvi,j])sin⁡[(ωi,j+Ω)t]}

    where δui,j and δvi,j are the components of δu corresponding to the respective harmonics ωi,j.

    We see that a motion of the harmonic variables at frequency Ω appears as motion of δxi(t) at frequencies ωi,j±Ω.

    To make sense of this, we normalize the vector δu and use normalised components δu^i,j and δv^i,j. We also define the Lorentzian distribution

    L(x)x0,γ=1(x−x0)2+γ2

    We see that all components of δxi(t) are proportional to L(Ω)Im[λ],Re[λ]. The first and last two summands are Lorentzians centered at ±Ω which oscillate at ωi,j±Ω, respectively. From this, we can extract the linear response function in Fourier space, χ(ω~)

    |χ[δxi](ω~)|2=∑j=1Mi{[(Re[δu^i,j]−Im[δv^i,j])2+(Im[δu^i,j]+Re[δv^i,j])2]L(ωi,j−ω~)Im[λ],Re[λ]+[(Re[δu^i,j]+Im[δv^i,j])2+(Re[δv^i,j]−Im[δu^i,j])2]L(ω~−ωi,j)Im[λ],Re[λ]}

    Keeping in mind that L(x)x0,γ=L(x+Δ)x0+Δ,γ and the normalization δu^i,j2+δv^i,j2=1, we can rewrite this as

    |χ[δxi](ω~)|2=∑j=1Mi(1+αi,j)L(ω~)ωi,j−Im[λ],Re[λ]+(1−αi,j)L(ω~)ωi,j+Im[λ],Re[λ]

    where

    αi,j=2(Im[δu^i,j]Re[δv^i,j]−Re[δu^i,j]Im[δv^i,j])

    The above solution applies to every eigenvalue λ of the Jacobian. It is now clear that the linear response function χ[δxi](ω~) contains for each eigenvalue λr and harmonic ωi,j :

    • A Lorentzian centered at ωi,j−Im[λr] with amplitude 1+αi,j(r)

    • A Lorentzian centered at ωi,j+Im[λr] with amplitude 1−αi,j(r)

    Sidenote: As J a real matrix, there is an eigenvalue λr∗ for each λr. The maximum number of peaks in the linear response is thus equal to the dimensionality of u(T).

    The linear response of the system in the state u0 is thus fully specified by the complex eigenvalues and eigenvectors of J(u0). In HarmonicBalance.jl, the module LinearResponse creates a set of plottable Lorentzian objects to represent this.

    Check out this example of the linear response module of HarmonicBalance.jl

    + + + + \ No newline at end of file diff --git a/v0.10.7/examples/index.html b/v0.10.7/examples/index.html new file mode 100644 index 00000000..e2072c4f --- /dev/null +++ b/v0.10.7/examples/index.html @@ -0,0 +1,27 @@ + + + + + + Examples | HarmonicBalance.jl + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/v0.10.7/examples/parametric_via_three_wave_mixing.html b/v0.10.7/examples/parametric_via_three_wave_mixing.html new file mode 100644 index 00000000..1d3cd41b --- /dev/null +++ b/v0.10.7/examples/parametric_via_three_wave_mixing.html @@ -0,0 +1,69 @@ + + + + + + Parametric Pumping via Three-Wave Mixing | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Parametric Pumping via Three-Wave Mixing ​

    julia
    using HarmonicBalance, Plots
+using Plots.Measures
+using Random

    System ​

    julia
    @variables β α ω ω0 F γ t x(t) # declare constant variables and a function x(t)
+diff_eq = DifferentialEquation(
+    d(x, t, 2) + ω0^2 * x + β * x^2 + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
+)
+add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)

    1st order Krylov expansion ​

    julia
    harmonic_eq = get_krylov_equations(diff_eq; order=1)
+harmonic_eq.equations

    If we both have quadratic and cubic nonlineariy, we observe the normal duffing oscillator response.

    julia
    varied = (ω => range(0.99, 1.1, 200)) # range of parameter values
+fixed = (α => 1.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025) # fixed parameters
+
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+plot(result; y="u1^2+v1^2")

    If we set the cubic nonlinearity to zero, we recover the driven damped harmonic oscillator. Indeed, thefirst order the quadratic nonlinearity has no affect on the system.

    julia
    varied = (ω => range(0.99, 1.1, 100))
+fixed = (α => 0.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025)
+
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+plot(result; y="u1^2+v1^2")

    2nd order Krylov expansion ​

    The quadratic nonlinearity β together with the drive at 2ω gives the effective parametric drive λeff=2F1β3mω2. But the cubic nonlinearity α is still needed to get the period doubling bifurcation through λeff.

    julia
    @variables β α ω ω0 F γ t x(t)
+diff_eq = DifferentialEquation(
+    d(x, t, 2) + ω0^2 * x + β * x^2 + α * x^3 + γ * d(x, t) ~ F * cos(2ω * t), x
+)
+
+add_harmonic!(diff_eq, x, ω)
+harmonic_eq2 = get_krylov_equations(diff_eq; order=2)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, ω0, F, β, α, γ
+
+Harmonic ansatz: 
+xˍt(t) = 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+(-(1//6)*F*v1(T)*β + (5//12)*(u1(T)^2)*v1(T)*(β^2) + (5//12)*(v1(T)^3)*(β^2) + (1//8)*v1(T)*(γ^2)*(ω^2) + (1//8)*v1(T)*(ω^4) - (1//4)*v1(T)*(ω^2)*(ω0^2) + (1//8)*v1(T)*(ω0^4) - (3//8)*(u1(T)^2)*v1(T)*α*(ω^2) + (3//8)*(u1(T)^2)*v1(T)*α*(ω0^2) - (3//8)*(v1(T)^3)*α*(ω^2) + (3//8)*(v1(T)^3)*α*(ω0^2) + (51//256)*(u1(T)^4)*v1(T)*(α^2) + (51//128)*(u1(T)^2)*(v1(T)^3)*(α^2) + (51//256)*(v1(T)^5)*(α^2)) / (ω^3) + (-(1//2)*u1(T)*γ*ω + (1//2)*v1(T)*(ω^2) - (1//2)*v1(T)*(ω0^2) - (3//8)*(u1(T)^2)*v1(T)*α - (3//8)*(v1(T)^3)*α) / ω ~ Differential(T)(u1(T))
+
+(-(1//6)*F*u1(T)*β - (5//12)*(u1(T)^3)*(β^2) - (5//12)*u1(T)*(v1(T)^2)*(β^2) - (1//8)*u1(T)*(γ^2)*(ω^2) - (1//8)*u1(T)*(ω^4) + (1//4)*u1(T)*(ω^2)*(ω0^2) - (1//8)*u1(T)*(ω0^4) + (3//8)*(u1(T)^3)*α*(ω^2) - (3//8)*(u1(T)^3)*α*(ω0^2) + (3//8)*u1(T)*(v1(T)^2)*α*(ω^2) - (3//8)*u1(T)*(v1(T)^2)*α*(ω0^2) - (51//256)*(u1(T)^5)*(α^2) - (51//128)*(u1(T)^3)*(v1(T)^2)*(α^2) - (51//256)*u1(T)*(v1(T)^4)*(α^2)) / (ω^3) + (-(1//2)*u1(T)*(ω^2) + (1//2)*u1(T)*(ω0^2) - (1//2)*v1(T)*γ*ω + (3//8)*(u1(T)^3)*α + (3//8)*u1(T)*(v1(T)^2)*α) / ω ~ Differential(T)(v1(T))
    julia
    varied = (ω => range(0.4, 1.1, 500))
+fixed = (α => 1.0, β => 2.0, ω0 => 1.0, γ => 0.001, F => 0.005)
+
+result = get_steady_states(harmonic_eq2, varied, fixed; threading=true)
+plot(result; y="v1")

    julia
    varied = (ω => range(0.4, 0.6, 100), F => range(1e-6, 0.01, 50))
+fixed = (α => 1.0, β => 2.0, ω0 => 1.0, γ => 0.01)
+
+result = get_steady_states(
+    harmonic_eq2, varied, fixed; threading=true, method=:total_degree
+)
+plot_phase_diagram(result; class="stable")

    This page was generated using Literate.jl.

    + + + + \ No newline at end of file diff --git a/v0.10.7/examples/parametron.html b/v0.10.7/examples/parametron.html new file mode 100644 index 00000000..dc5773d2 --- /dev/null +++ b/v0.10.7/examples/parametron.html @@ -0,0 +1,116 @@ + + + + + + Parametrically driven resonator | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Parametrically driven resonator ​

    One of the most famous effects displaced by nonlinear oscillators is parametric resonance, where the frequency of the linear resonator is modulated in time Phys. Rev. E 94, 022201 (2016). In the following we analyse this system, governed by the equations

    x¨(t)+γx˙(t)+Ω2(1−λcos⁡(2ωt+ψ))x+αx3+ηx2x˙+Fd(t)=0

    where for completeness we also considered an external drive term Fd(t)=Fcos⁡(ωt+θ) and a nonlinear damping term ηx2x˙

    To implement this system in Harmonic Balance, we first import the library

    julia
    using HarmonicBalance

    Subsequently, we type define parameters in the problem and the oscillating amplitude function x(t) using the variables macro from Symbolics.jl

    julia
    @variables ω₀ γ λ F η α ω t x(t)
+
+natural_equation =
+    d(d(x, t), t) +
+    γ * d(x, t) +
+    (ω₀^2 - λ * cos(2 * ω * t)) * x +
+    α * x^3 +
+    η * d(x, t) * x^2
+forces = F * cos(ω * t)
+diff_eq = DifferentialEquation(natural_equation + forces, x)
    System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ;   
+
+Differential(t)(Differential(t)(x(t))) + F*cos(t*ω) + Differential(t)(x(t))*γ + x(t)*(-cos(2t*ω)*λ + ω₀^2) + (x(t)^3)*α + (x(t)^2)*Differential(t)(x(t))*η ~ 0

    Note that an equation of the form

    mx¨+mω02(1−λcos⁡(2ωt+ψ))x+γx˙+αx3+ηx2x˙=Fcos⁡ωt

    can be brought to dimensionless form by rescaling the units as described in Phys. Rev. E 94, 022201 (2016).

    We are interested in studying the response of the oscillator to parametric driving and forcing. In particular, we focus on the first parametric resonance of the system, i.e. operating around twice the bare frequency of the undriven oscillator ω while the frequency of the external drive is also ω. For this purpose, we consider a harmonic ansatz which contains a single frequency: x(t)≈ucos⁡(ωt)+vsin⁡(ωt). In HarmonicBalance, we can do this via add_harmonic command:

    julia
    add_harmonic!(diff_eq, x, ω);

    and replacing this by the time independent (averaged) equations of motion. This can be simply done by writing

    julia
    harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω₀, λ, F, η
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+F - (1//2)*u1(T)*λ + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω₀^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*Differential(T)(u1(T))*η + (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*η + (3//4)*u1(T)*(v1(T)^2)*α + (1//4)*(v1(T)^2)*Differential(T)(u1(T))*η + (1//4)*(u1(T)^2)*v1(T)*η*ω + (1//4)*(v1(T)^3)*η*ω ~ 0
+
+Differential(T)(v1(T))*γ + (1//2)*v1(T)*λ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω₀^2) + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*η + (3//4)*(u1(T)^2)*v1(T)*α + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*η + (3//4)*Differential(T)(v1(T))*(v1(T)^2)*η + (3//4)*(v1(T)^3)*α - (1//4)*(u1(T)^3)*η*ω - (1//4)*u1(T)*(v1(T)^2)*η*ω ~ 0

    The output of these equations are consistent with the result found in the literature. Now we are interested in the linear response spectrum, which we can obtain from the solutions to the averaged equations (rotating frame) as a function of the external drive, after fixing all other parameters in the system. A call to get_steady_states then retrieves all steadystates found allong the sweep employing the homotopy continuation method, which occurs in a complex space (see the nice HomotopyContinuation.jl docs)

    1D parameters ​

    We start with a varied set containing one parameter, ω,

    julia
    fixed = (ω₀ => 1.0, γ => 1e-2, λ => 5e-2, F => 1e-3, α => 1.0, η => 0.3)
+varied = ω => range(0.9, 1.1, 100)
+
+result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 100 parameter points
+
+Solution branches:   5
+   of which real:    5
+   of which stable:  3
+
+Classes: stable, physical, Hopf, binary_labels

    In get_steady_states, the default value for the keyword method=:random_warmup initiates the homotopy in a generalised version of the harmonic equations, where parameters become random complex numbers. A parameter homotopy then follows to each of the frequency values ω in sweep. This offers speed-up, but requires to be tested in each scenario againts the method :total_degree, which initializes the homotopy in a total degree system (maximum number of roots), but needs to track significantly more homotopy paths and there is slower. The threading keyword enables parallel tracking of homotopy paths, and it's set to false simply because we are using a single core computer for now.

    After solving the system, we can save the full output of the simulation and the model (e.g. symbolic expressions for the harmonic equations) into a file

    julia
    HarmonicBalance.save("parametron_result.jld2", result);
    ┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(cos).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(identity).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631
+┌ Warning: Attempting to store typeof(identity).
+│  Function types cannot be propertly stored in JLD2 files.
+│  Loading may yield unexpected results.
+└ @ JLD2 ~/.julia/packages/JLD2/JHhTf/src/data/writing_datatypes.jl:631

    During the execution of get_steady_states, different solution branches are classified by their proximity in complex space, with subsequent filtering of real (physically accceptable solutions). In addition, the stability properties of each steady state is assesed from the eigenvalues of the Jacobian matrix. All this information can be succintly represented in a 1D plot via

    julia
    plot(result; x="ω", y="sqrt(u1^2 + v1^2)")

    The user can also introduce custom clases based on parameter conditions via classify_solutions!. Plots can be overlaid and use keywords from Plots, MarkdownAST.LineBreak()

    julia
    classify_solutions!(result, "sqrt(u1^2 + v1^2) > 0.1", "large")
+plot(result, "sqrt(u1^2 + v1^2)"; class=["physical", "large"], style=:dash)
+plot!(result, "sqrt(u1^2 + v1^2)"; not_class="large")

    Alternatively, we may visualise all underlying solutions, including complex ones,

    julia
    plot(result, "sqrt(u1^2 + v1^2)"; class="all")

    2D parameters ​

    The parametrically driven oscillator boasts a stability diagram called "Arnold's tongues" delineating zones where the oscillator is stable from those where it is exponentially unstable (if the nonlinearity was absence). We can retrieve this diagram by calculating the steady states as a function of external detuning δ=ωL−ω0 and the parametric drive strength λ.

    To perform a 2D sweep over driving frequency ω and parametric drive strength λ, we keep fixed from before but include 2 variables in varied

    julia
    varied = (ω => range(0.8, 1.2, 50), λ => range(0.001, 0.6, 50))
+result_2D = get_steady_states(harmonic_eq, varied, fixed);
    
+Solving for 2500 parameters...  53%|██████████▋         |  ETA: 0:00:01
+  # parameters solved:  1328
+  # paths tracked:      6640
+
+
+
+
+
+
+Solving for 2500 parameters...  83%|████████████████▋   |  ETA: 0:00:00
+  # parameters solved:  2085
+  # paths tracked:      10425
+
+
+
+
+
+
+Solving for 2500 parameters... 100%|████████████████████| Time: 0:00:01
+  # parameters solved:  2500
+  # paths tracked:      12500

    Now, we count the number of solutions for each point and represent the corresponding phase diagram in parameter space. This is done using plot_phase_diagram. Only counting stable solutions,

    julia
    plot_phase_diagram(result_2D; class="stable")

    In addition to phase diagrams, we can plot functions of the solution. The syntax is identical to 1D plotting. Let us overlay 2 branches into a single plot,

    julia
    # overlay branches with different colors
+plot(result_2D, "sqrt(u1^2 + v1^2)"; branch=1, class="stable", camera=(60, -40))
+plot!(result_2D, "sqrt(u1^2 + v1^2)"; branch=2, class="stable", color=:red)

    Note that solutions are ordered in parameter space according to their closest neighbors. Plots can again be limited to a given class (e.g stable solutions only) through the keyword argument class.

    This page was generated using Literate.jl.

    + + + + \ No newline at end of file diff --git a/v0.10.7/examples/wave_mixing.html b/v0.10.7/examples/wave_mixing.html new file mode 100644 index 00000000..dd8db0eb --- /dev/null +++ b/v0.10.7/examples/wave_mixing.html @@ -0,0 +1,79 @@ + + + + + + Three Wave Mixing vs four wave mixing | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Three Wave Mixing vs four wave mixing ​

    Packages ​

    We load the following packages into our environment:

    julia
    using HarmonicBalance, Plots
+using Plots.Measures
+using Random
+
+Random.seed!(1234);
+Threads.nthreads() # check number of threads
    1

    system ​

    julia
    @variables β α ω ω0 F γ t x(t) # declare constant variables and a function x(t)
+diff_eq = DifferentialEquation(
+    d(x, t, 2) + ω0^2 * x + β * x^2 + α * x^3 + γ * d(x, t) ~ F * cos(ω * t), x
+)
+
+add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)
+add_harmonic!(diff_eq, x, 2ω)
+add_harmonic!(diff_eq, x, 3ω)
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 6 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T), u3(T), v3(T)
+Parameters: ω, α, γ, ω0, β, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt) + u2(T)*cos(2ωt) + v2(T)*sin(2ωt) + u3(T)*cos(3ωt) + v3(T)*sin(3ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ + u1(T)*u2(T)*β - u1(T)*(ω^2) + u1(T)*(ω0^2) + v3(T)*v2(T)*β + v2(T)*v1(T)*β + v1(T)*γ*ω + u3(T)*u2(T)*β + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*u3(T)*α + (3//2)*u1(T)*(v3(T)^2)*α + (3//2)*u1(T)*v3(T)*v1(T)*α + (3//2)*u1(T)*(v2(T)^2)*α + (3//4)*u1(T)*(v1(T)^2)*α + (3//2)*u1(T)*(u3(T)^2)*α + (3//2)*u1(T)*(u2(T)^2)*α + (3//2)*v3(T)*v2(T)*u2(T)*α - (3//4)*(v2(T)^2)*u3(T)*α - (3//4)*(v1(T)^2)*u3(T)*α + (3//4)*u3(T)*(u2(T)^2)*α ~ F
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω + u1(T)*v2(T)*β - u1(T)*γ*ω + v3(T)*u2(T)*β - v2(T)*u3(T)*β - v1(T)*u2(T)*β - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v3(T)*α + (3//4)*(u1(T)^2)*v1(T)*α - (3//2)*u1(T)*v1(T)*u3(T)*α + (3//2)*(v3(T)^2)*v1(T)*α + (3//4)*v3(T)*(v2(T)^2)*α - (3//4)*v3(T)*(v1(T)^2)*α - (3//4)*v3(T)*(u2(T)^2)*α + (3//2)*(v2(T)^2)*v1(T)*α + (3//2)*v2(T)*u3(T)*u2(T)*α + (3//4)*(v1(T)^3)*α + (3//2)*v1(T)*(u3(T)^2)*α + (3//2)*v1(T)*(u2(T)^2)*α ~ 0
+
+Differential(T)(u2(T))*γ + (4//1)*Differential(T)(v2(T))*ω + (1//2)*(u1(T)^2)*β + u1(T)*u3(T)*β + v3(T)*v1(T)*β + (2//1)*v2(T)*γ*ω - (1//2)*(v1(T)^2)*β - (4//1)*u2(T)*(ω^2) + u2(T)*(ω0^2) + (3//2)*(u1(T)^2)*u2(T)*α + (3//2)*u1(T)*v3(T)*v2(T)*α + (3//2)*u1(T)*u3(T)*u2(T)*α + (3//2)*(v3(T)^2)*u2(T)*α - (3//2)*v3(T)*v1(T)*u2(T)*α + (3//4)*(v2(T)^2)*u2(T)*α + (3//2)*v2(T)*v1(T)*u3(T)*α + (3//2)*(v1(T)^2)*u2(T)*α + (3//2)*(u3(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0//1
+
+-(4//1)*Differential(T)(u2(T))*ω + Differential(T)(v2(T))*γ + u1(T)*v3(T)*β + u1(T)*v1(T)*β - (4//1)*v2(T)*(ω^2) + v2(T)*(ω0^2) - v1(T)*u3(T)*β - (2//1)*u2(T)*γ*ω + (3//2)*(u1(T)^2)*v2(T)*α + (3//2)*u1(T)*v3(T)*u2(T)*α - (3//2)*u1(T)*v2(T)*u3(T)*α + (3//2)*(v3(T)^2)*v2(T)*α + (3//2)*v3(T)*v2(T)*v1(T)*α + (3//4)*(v2(T)^3)*α + (3//2)*v2(T)*(v1(T)^2)*α + (3//2)*v2(T)*(u3(T)^2)*α + (3//4)*v2(T)*(u2(T)^2)*α + (3//2)*v1(T)*u3(T)*u2(T)*α ~ 0//1
+
+Differential(T)(u3(T))*γ + (6//1)*Differential(T)(v3(T))*ω + u1(T)*u2(T)*β + (3//1)*v3(T)*γ*ω - v2(T)*v1(T)*β - (9//1)*u3(T)*(ω^2) + u3(T)*(ω0^2) + (1//4)*(u1(T)^3)*α + (3//2)*(u1(T)^2)*u3(T)*α - (3//4)*u1(T)*(v2(T)^2)*α - (3//4)*u1(T)*(v1(T)^2)*α + (3//4)*u1(T)*(u2(T)^2)*α + (3//4)*(v3(T)^2)*u3(T)*α + (3//2)*(v2(T)^2)*u3(T)*α + (3//2)*v2(T)*v1(T)*u2(T)*α + (3//2)*(v1(T)^2)*u3(T)*α + (3//4)*(u3(T)^3)*α + (3//2)*u3(T)*(u2(T)^2)*α ~ 0//1
+
+-(6//1)*Differential(T)(u3(T))*ω + Differential(T)(v3(T))*γ + u1(T)*v2(T)*β - (9//1)*v3(T)*(ω^2) + v3(T)*(ω0^2) + v1(T)*u2(T)*β - (3//1)*u3(T)*γ*ω + (3//2)*(u1(T)^2)*v3(T)*α + (3//4)*(u1(T)^2)*v1(T)*α + (3//2)*u1(T)*v2(T)*u2(T)*α + (3//4)*(v3(T)^3)*α + (3//2)*v3(T)*(v2(T)^2)*α + (3//2)*v3(T)*(v1(T)^2)*α + (3//4)*v3(T)*(u3(T)^2)*α + (3//2)*v3(T)*(u2(T)^2)*α + (3//4)*(v2(T)^2)*v1(T)*α - (1//4)*(v1(T)^3)*α - (3//4)*v1(T)*(u2(T)^2)*α ~ 0//1

    four wave mixing ​

    If we only have a cubic nonlineariy α, we observe the normal duffing oscillator response with no response at 2ω.

    julia
    varied = (ω => range(0.9, 1.2, 200)) # range of parameter values
+fixed = (α => 1.0, β => 0.0, ω0 => 1.0, γ => 0.005, F => 0.0025) # fixed parameters
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)# compute steady states
+
+p1 = plot(result; y="√(u1^2+v1^2)", legend=:best)
+p2 = plot(result; y="√(u2^2+v2^2)", legend=:best, ylims=(-0.1, 0.1))
+p3 = plot(result; y="√(u3^2+v3^2)", legend=:best)
+plot(p1, p2, p3; layout=(1, 3), size=(900, 300), margin=5mm)

    Three wave mixing ​

    If we only have a cubic nonlineariy α, we observe the normal duffing oscillator response with no response at 2ω.

    We would like to investigate the three-wave mixing of the driven Duffing oscillator. This means we can excite the system resonantly if the oscillation frequencies ω1 and ω2 fullfil the conditions ω1±ω2=±ω0. Here, we will especially focus on the degenerate three wave mixing, where ω2=ω0 such that 2ω0=ω1. This is a very important process in quantum optics, since it allows us to generate photons with a frequency in the visible range from photons with a frequency in the infrared range. This is called frequency doubling and is used in many applications, e.g. in laser pointers.

    julia
    varied = (ω => range(0.9, 1.2, 200))
+fixed = (α => 0.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025)
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+
+p1 = plot(result; y="√(u1^2+v1^2)", legend=:best)
+p2 = plot(result; y="√(u2^2+v2^2)", legend=:best, ylims=(-0.1, 0.1))
+p3 = plot(result; y="√(u3^2+v3^2)", legend=:best)
+plot(p1, p2, p3; layout=(1, 3), size=(900, 300), margin=5mm)

    Both ​

    If we only have a cubic nonlineariy α, we observe the normal duffing oscillator response with no response at 2ω.

    We would like to investigate the three-wave mixing of the driven Duffing oscillator. This means we can excite the system resonantly if the oscillation frequencies ω1 and ω2 fullfil the conditions ω1±ω2=±ω0. Here, we will especially focus on the degenerate three wave mixing, where ω2=ω0 such that 2ω0=ω1. This is a very important process in quantum optics, since it allows us to generate photons with a frequency in the visible range from photons with a frequency in the infrared range. This is called frequency doubling and is used in many applications, e.g. in laser pointers.

    julia
    varied = (ω => range(0.9, 1.2, 200))
+fixed = (α => 1.0, β => 1.0, ω0 => 1.0, γ => 0.005, F => 0.0025)
+result = get_steady_states(harmonic_eq, varied, fixed; threading=true)
+
+p1 = plot(result; y="√(u1^2+v1^2)", legend=:best)
+p2 = plot(result; y="√(u2^2+v2^2)", legend=:best, ylims=(-0.1, 0.1))
+p3 = plot(result; y="√(u3^2+v3^2)", legend=:best)
+plot(p1, p2, p3; layout=(1, 3), size=(900, 300), margin=5mm)

    This page was generated using Literate.jl.

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z@DLDAL!2+(K0to!ewuZ~ZeEG>j1V9M!UqBP0RJnXK7i4?F??D_k`e-h01pB90PF~e z|0efyyz=N7AwUR(4+7}_uM33#*cV`N__U5BB?Jfo9s+Sj?-POWANpcCuRMB22oM6{ zg8+Pa<_E%m#Qqr*KCL532?0WYhXB?9O%8 + + + + + HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    HarmonicBalance.jl

    Efficient Floquet expansions for nonlinear driven systems

    A Julia suite for nonlinear dynamics using harmonic balance

    Getting Started
    Tutorials
    View on GitHub
    HarmonicBalance.jl

    Non-equilibrium steady states

    Compute all stationary states in a one or two-dimensional parameter sweep.

    Linear Response

    Explore the fluctuations on top of the non-equilibrium steady states.

    Limit Cycles

    Find limit cycles involving many frequencies.

    + + + + \ No newline at end of file diff --git a/v0.10.7/introduction/citation.html b/v0.10.7/introduction/citation.html new file mode 100644 index 00000000..21f092b8 --- /dev/null +++ b/v0.10.7/introduction/citation.html @@ -0,0 +1,26 @@ + + + + + + Citation | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Citation ​

    If you use HarmonicBalance.jl in your project, we kindly ask you to cite this paper, namely:

    HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance, Jan Košata, Javier del Pino, Toni L. Heugel, Oded Zilberberg, SciPost Phys. Codebases 6 (2022)

    The limit cycle finding algorithm is based on the work of this paper:

    Limit cycles as stationary states of an extended harmonic balance ansatz J. del Pino, J. Košata, and O. Zilberberg, Phys. Rev. Res. 6, 033180 (2024).

    + + + + \ No newline at end of file diff --git a/v0.10.7/introduction/index.html b/v0.10.7/introduction/index.html new file mode 100644 index 00000000..b3d06d5a --- /dev/null +++ b/v0.10.7/introduction/index.html @@ -0,0 +1,46 @@ + + + + + + Installation | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Installation ​

    It is easy to install HarmonicBalance.jl as we are registered in the Julia General registry. You can simply run the following command in the Julia REPL:

    julia
    julia> using Pkg
+julia> Pkg.add("HarmonicBalance")

    or

    julia
    julia> ] # `]` should be pressed
+julia> Pkg.add("HarmonicBalance")

    You can check which version you have installled with the command

    julia
    julia> ]
+julia> status HarmonicBalance

    Getting Started ​

    Let us find the steady states of an external driven Duffing oscillator with nonlinear damping. Its equation of motion is:

    x¨(t)+γx˙(t)+ω02x(t)⏟damped harmonic oscillator+αx(t)3⏟Duffing coefficient=Fcos⁡(ωt)⏟periodic drive
    julia
    using HarmonicBalance
+@variables α ω ω0 F t η x(t) # declare constant variables and a function x(t)
+eom = d(x,t,2) + ω0^2*x + α*x^3 + η*d(x,t)*x^2 ~ F*cos(ω*t)
+diff_eq = DifferentialEquation(eom, x)
+add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)
+
+# implement ansatz to get harmonic equations
+harmonic_eq = get_harmonic_equations(diff_eq)
+
+fixed = (α => 1.0, ω0 => 1.0, F => 0.01, η => 0.1)   # fixed parameters
+varied = ω => range(0.9, 1.2, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 100 parameter points
+
+Solution branches:   3
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels

    The obtained steady states can be plotted as a function of the driving frequency:

    julia
    plot(result, "sqrt(u1^2 + v1^2)")

    If you want learn more on what you can do with HarmonicBalance.jl, check out the tutorials. We also have collected some examples of different physical systems.

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zy>#5NRCyvufinLH_?CPcD`l3u7-#(v!{*MPl&e#=FgsnL(`a51EpYbg6UrSM&o^w# Rh6UJAvQmnYmExZR{~tF|-gp22 literal 0 HcmV?d00001 diff --git a/v0.10.7/manual/Krylov-Bogoliubov_method.html b/v0.10.7/manual/Krylov-Bogoliubov_method.html new file mode 100644 index 00000000..87eeab18 --- /dev/null +++ b/v0.10.7/manual/Krylov-Bogoliubov_method.html @@ -0,0 +1,54 @@ + + + + + + Krylov-Bogoliubov Averaging Method | HarmonicBalance.jl + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/v0.10.7/manual/entering_eom.html b/v0.10.7/manual/entering_eom.html new file mode 100644 index 00000000..f1a1b95f --- /dev/null +++ b/v0.10.7/manual/entering_eom.html @@ -0,0 +1,43 @@ + + + + + + Entering equations of motion | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Entering equations of motion ​

    The struct DifferentialEquation is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using add_harmonic!.

    DifferentialEquation.equations stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.

    # HarmonicBalance.DifferentialEquation — Type.
    julia
    mutable struct DifferentialEquation

    Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for HarmonicBalance.jl ; after inputting the equations, the harmonics ansatz needs to be specified using add_harmonic!.

    Fields

    • equations::OrderedCollections.OrderedDict{Num, Equation}: Assigns to each variable an equation of motion.

    • harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}: Assigns to each variable a set of harmonics.

    Example

    julia
    julia> @variables t, x(t), y(t), ω0, ω, F, k;
+
+# equivalent ways to enter the simple harmonic oscillator
+julia> DifferentialEquation(d(x,t,2) + ω0^2 * x - F * cos(ω*t), x);
+julia> DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x);
+
+# two coupled oscillators, one of them driven
+julia> DifferentialEquation([d(x,t,2) + ω0^2 * x - k*y, d(y,t,2) + ω0^2 * y - k*x] .~ [F * cos(ω*t), 0], [x,y]);

    source


    # HarmonicBalance.add_harmonic! — Function.
    julia
    add_harmonic!(diff_eom::DifferentialEquation, var::Num, ω)

    Add the harmonic ω to the harmonic ansatz used to expand the variable var in diff_eom.

    Example

    define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω

    julia
    julia> @variables t, x(t), y(t), ω0, ω, F, k;
+julia> diff_eq = DifferentialEquation(d(x,t,2) + ω0^2 * x ~ F * cos(ω*t), x);
+julia> add_harmonic!(diff_eq, x, ω) # expand x using ω
+
+System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ω;
+
+(ω0^2)*x(t) + Differential(t)(Differential(t)(x(t))) ~ F*cos(t*ω)

    source


    # Symbolics.get_variables — Method.
    julia
    get_variables(diff_eom::DifferentialEquation) -> Vector{Num}

    Return the dependent variables of diff_eom.

    source


    # HarmonicBalance.get_independent_variables — Method.
    julia
    get_independent_variables(
+    diff_eom::DifferentialEquation
+) -> Any

    Return the independent dependent variables of diff_eom.

    source


    + + + + \ No newline at end of file diff --git a/v0.10.7/manual/extracting_harmonics.html b/v0.10.7/manual/extracting_harmonics.html new file mode 100644 index 00000000..403c23bb --- /dev/null +++ b/v0.10.7/manual/extracting_harmonics.html @@ -0,0 +1,57 @@ + + + + + + Extracting harmonic equations | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Extracting harmonic equations ​

    Harmonic Balance method ​

    Once a DifferentialEquation is defined and its harmonics specified, one can extract the harmonic equations using get_harmonic_equations, which itself is composed of the subroutines harmonic_ansatz, slow_flow, fourier_transform! and drop_powers.

    The harmonic equations use an additional time variable specified as slow_time in get_harmonic_equations. This is essentially a label distinguishing the time dependence of the harmonic variables (expected to be slow) from that of the oscillating terms (expeted to be fast). When the equations are Fourier-transformed to remove oscillating terms, slow_time is treated as a constant. Such an approach is exact when looking for steady states.

    # HarmonicBalance.get_harmonic_equations — Function.
    julia
    get_harmonic_equations(diff_eom::DifferentialEquation; fast_time=nothing, slow_time=nothing)

    Apply the harmonic ansatz, followed by the slow-flow, Fourier transform and dropping higher-order derivatives to obtain a set of ODEs (the harmonic equations) governing the harmonics of diff_eom.

    The harmonics evolve in slow_time, the oscillating terms themselves in fast_time. If no input is used, a variable T is defined for slow_time and fast_time is taken as the independent variable of diff_eom.

    By default, all products of order > 1 of slow_time-derivatives are dropped, which means the equations are linear in the time-derivatives.

    Example

    julia
    julia> @variables t, x(t), ω0, ω, F;
+
+# enter the simple harmonic oscillator
+julia> diff_eom = DifferentialEquation( d(x,t,2) + ω0^2 * x ~ F *cos(ω*t), x);
+
+# expand x in the harmonic ω
+julia> add_harmonic!(diff_eom, x, ω);
+
+# get equations for the harmonics evolving in the slow time T
+julia> harmonic_eom = get_harmonic_equations(diff_eom)
+
+A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω0, ω, F
+
+Harmonic ansatz:
+x(t) = u1*cos(ωt) + v1*sin(ωt)
+
+Harmonic equations:
+
+(ω0^2)*u1(T) + (2//1)*ω*Differential(T)(v1(T)) - (ω^2)*u1(T) ~ F
+
+(ω0^2)*v1(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) ~ 0

    source


    # HarmonicBalance.harmonic_ansatz — Function.
    julia
    harmonic_ansatz(eom::DifferentialEquation, time::Num; coordinates="Cartesian")

    Expand each variable of diff_eom using the harmonics assigned to it with time as the time variable. For each harmonic of each variable, instance(s) of HarmonicVariable are automatically created and named.

    source


    # HarmonicBalance.slow_flow — Function.
    julia
    slow_flow(eom::HarmonicEquation; fast_time::Num, slow_time::Num, degree=2)

    Removes all derivatives w.r.t fast_time (and their products) in eom of power degree. In the remaining derivatives, fast_time is replaced by slow_time.

    source


    # HarmonicBalance.fourier_transform — Function.
    julia
    fourier_transform(
+    eom::HarmonicEquation,
+    time::Num
+) -> HarmonicEquation

    Extract the Fourier components of eom corresponding to the harmonics specified in eom.variables. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated time does not appear in the resulting equations anymore.

    Underlying assumption: all time-dependences are harmonic.

    source


    # HarmonicBalance.drop_powers — Function.
    julia
    drop_powers(expr, vars, deg)

    Remove parts of expr where the combined power of vars is => deg.

    Example

    julia
    julia> @variables x,y;
+julia>drop_powers((x+y)^2, x, 2)
+y^2 + 2*x*y
+julia>drop_powers((x+y)^2, [x,y], 2)
+0
+julia>drop_powers((x+y)^2 + (x+y)^3, [x,y], 3)
+x^2 + y^2 + 2*x*y

    source


    HarmonicVariable and HarmonicEquation types ​

    The equations governing the harmonics are stored using the two following structs. When going from the original to the harmonic equations, the harmonic ansatz xi(t)=∑j=1Mui,j(T)cos⁡(ωi,jt)+vi,j(T)sin⁡(ωi,jt) is used. Internally, each pair (ui,j,vi,j) is stored as a HarmonicVariable. This includes the identification of ωi,j and xi(t), which is needed to later reconstruct xi(t).

    # HarmonicBalance.HarmonicVariable — Type.
    julia
    mutable struct HarmonicVariable

    Holds a variable stored under symbol describing the harmonic ω of natural_variable.

    Fields

    • symbol::Num: Symbol of the variable in the HarmonicBalance namespace.

    • name::String: Human-readable labels of the variable, used for plotting.

    • type::String: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)

    • ω::Num: The harmonic being described.

    • natural_variable::Num: The natural variable whose harmonic is being described.

    source


    When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a HarmonicEquation. For an initial equation of motion consisting of M variables, each expanded in N harmonics, the resulting HarmonicEquation holds 2NM equations of 2NM variables. Each symbol not corresponding to a variable is identified as a parameter.

    A HarmonicEquation can be either parsed into a steady-state Problem or solved using a dynamical ODE solver.

    # HarmonicBalance.HarmonicEquation — Type.
    julia
    mutable struct HarmonicEquation

    Holds a set of algebraic equations governing the harmonics of a DifferentialEquation.

    Fields

    • equations::Vector{Equation}: A set of equations governing the harmonics.

    • variables::Vector{HarmonicVariable}: A set of variables describing the harmonics.

    • parameters::Vector{Num}: The parameters of the equation set.

    • natural_equation::DifferentialEquation: The natural equation (before the harmonic ansatz was used).

    source


    + + + + \ No newline at end of file diff --git a/v0.10.7/manual/linear_response.html b/v0.10.7/manual/linear_response.html new file mode 100644 index 00000000..e66b4821 --- /dev/null +++ b/v0.10.7/manual/linear_response.html @@ -0,0 +1,30 @@ + + + + + + Linear response (WIP) | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Linear response (WIP) ​

    This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.

    The methodology used is explained in Jan Kosata phd thesis.

    Stability ​

    The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,

    # HarmonicBalance.LinearResponse.get_Jacobian — Function.
    julia
    get_Jacobian(eom)

    Obtain the symbolic Jacobian matrix of eom (either a HarmonicEquation or a DifferentialEquation). This is the linearised left-hand side of F(u) = du/dT.

    source

    Obtain a Jacobian from a DifferentialEquation by first converting it into a HarmonicEquation.

    source

    Get the Jacobian of a set of equations eqs with respect to the variables vars.

    source


    Linear response ​

    The response to white noise can be shown with plot_linear_response. Depending on the order argument, different methods are used.

    # HarmonicBalance.LinearResponse.plot_linear_response — Function.
    julia
    plot_linear_response(res::Result, nat_var::Num; Ω_range, branch::Int, order=1, logscale=false, show_progress=true, kwargs...)

    Plot the linear response to white noise of the variable nat_var for Result res on branch for input frequencies Ω_range. Slow-time derivatives up to order are kept in the process.

    Any kwargs are fed to Plots' gr().

    Solutions not belonging to the physical class are ignored.

    source


    First order ​

    The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues λ describes a Lorentzian peak in the response; Re[λ] gives its center and Im[λ] its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment.

    The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).

    Behind the scenes, the spectra are stored using the dedicated structs Lorentzian and JacobianSpectrum.

    # HarmonicBalance.LinearResponse.JacobianSpectrum — Type.
    julia
    mutable struct JacobianSpectrum

    Holds a set of Lorentzian objects belonging to a variable.

    Fields

    • peaks::Vector{HarmonicBalance.LinearResponse.Lorentzian}

    Constructor

    julia
    JacobianSpectrum(res::Result; index::Int, branch::Int)

    source


    # HarmonicBalance.LinearResponse.Lorentzian — Type.
    julia
    struct Lorentzian

    Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).

    Fields

    • ω0::Float64

    • Γ::Float64

    • A::Float64

    source


    Higher orders ​

    Setting order > 1 increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.

    # HarmonicBalance.LinearResponse.ResponseMatrix — Type.
    julia
    struct ResponseMatrix

    Holds the compiled response matrix of a system.

    Fields

    • matrix::Matrix{Function}: The response matrix (compiled).

    • symbols::Vector{Num}: Any symbolic variables in matrix to be substituted at evaluation.

    • variables::Vector{HarmonicVariable}: The frequencies of the harmonic variables underlying matrix. These are needed to transform the harmonic variables to the non-rotating frame.

    source


    # HarmonicBalance.LinearResponse.get_response — Function.
    julia
    get_response(
+    rmat::HarmonicBalance.LinearResponse.ResponseMatrix,
+    s::OrderedCollections.OrderedDict{Num, ComplexF64},
+    Ω
+) -> Any

    For rmat and a solution dictionary s, calculate the total response to a perturbative force at frequency Ω.

    source


    # HarmonicBalance.LinearResponse.get_response_matrix — Function.
    julia
    get_response_matrix(diff_eq::DifferentialEquation, freq::Num; order=2)

    Obtain the symbolic linear response matrix of a diff_eq corresponding to a perturbation frequency freq. This routine cannot accept a HarmonicEquation since there, some time-derivatives are already dropped. order denotes the highest differential order to be considered.

    source


    + + + + \ No newline at end of file diff --git a/v0.10.7/manual/plotting.html b/v0.10.7/manual/plotting.html new file mode 100644 index 00000000..5253bea5 --- /dev/null +++ b/v0.10.7/manual/plotting.html @@ -0,0 +1,35 @@ + + + + + + Analysis and plotting | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Analysis and plotting ​

    The key method for visualization is transform_solutions, which parses a string into a symbolic expression and evaluates it for every steady state solution.

    # HarmonicBalance.transform_solutions — Function.
    julia
    transform_solutions(
+    res::Result,
+    func;
+    branches,
+    realify
+) -> Vector

    Takes a Result object and a string f representing a Symbolics.jl expression. Returns an array with the values of f evaluated for the respective solutions. Additional substitution rules can be specified in rules in the format ("a" => val) or (a => val)

    source


    Plotting solutions ​

    The function plot is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by sort_solutions.

    # RecipesBase.plot — Method.
    julia
    plot(res::Result, varargs...; cut, kwargs...) -> Plots.Plot

    Plot a Result object.

    Class selection done by passing String or Vector{String} as kwarg:

    class       :   only plot solutions in this class(es) ("all" --> plot everything)
+not_class   :   do not plot solutions in this class(es)

    Other kwargs are passed onto Plots.gr().

    See also plot!

    The x,y,z arguments are Strings compatible with Symbolics.jl, e.g., y=2*sqrt(u1^2+v1^2) plots the amplitude of the first quadratures multiplied by 2.

    1D plots

    plot(res::Result; x::String, y::String, class="default", not_class=[], kwargs...)
+plot(res::Result, y::String; kwargs...) # take x automatically from Result

    Default behaviour is to plot stable solutions as full lines, unstable as dashed.

    If a sweep in two parameters were done, i.e., dim(res)==2, a one dimensional cut can be plotted by using the keyword cut were it takes a Pair{Num, Float64} type entry. For example, plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2)) plots a cut at λ = 0.2.

    2D plots

    plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)

    To make the 2d plot less chaotic it is required to specify the specific branch to plot, labeled by a Int64.

    The x and y axes are taken automatically from res

    source


    Plotting phase diagrams ​

    In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. plot_phase_diagram handles this for 1D and 2D datasets.

    # HarmonicBalance.plot_phase_diagram — Function.
    julia
    plot_phase_diagram(res::Result; kwargs...) -> Plots.Plot

    Plot the number of solutions in a Result object as a function of the parameters. Works with 1D and 2D datasets.

    Class selection done by passing String or Vector{String} as kwarg:

    class::String       :   only count solutions in this class ("all" --> plot everything)
+not_class::String   :   do not count solutions in this class

    Other kwargs are passed onto Plots.gr()

    source


    Plot spaghetti plot ​

    Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with plot_spaghetti.

    # HarmonicBalance.plot_spaghetti — Function.
    julia
    plot_spaghetti(res::Result; x, y, z, kwargs...)

    Plot a three dimension line plot of a Result object as a function of the parameters. Works with 1D and 2D datasets.

    Class selection done by passing String or Vector{String} as kwarg:

    class::String       :   only count solutions in this class ("all" --> plot everything)
+not_class::String   :   do not count solutions in this class

    Other kwargs are passed onto Plots.gr()

    source


    + + + + \ No newline at end of file diff --git a/v0.10.7/manual/saving.html b/v0.10.7/manual/saving.html new file mode 100644 index 00000000..95a927d5 --- /dev/null +++ b/v0.10.7/manual/saving.html @@ -0,0 +1,26 @@ + + + + + + Saving and loading | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Saving and loading ​

    All of the types native to HarmonicBalance.jl can be saved into a .jld2 file using save and loaded using load. Most of the saving/loading is performed using the package JLD2.jl, with the addition of reinstating the symbolic variables in the HarmonicBalance namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as Result can be saved and loaded with no loss of information.

    The function export_csv saves a .csv file which can be plot elsewhere.

    # HarmonicBalance.save — Function.
    julia
    save(filename, object)

    Saves object into .jld2 file filename (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.

    source


    # HarmonicBalance.load — Function.
    julia
    load(filename)

    Loads an object from filename. For objects containing symbolic expressions such as HarmonicEquation, the symbolic variables are reinstated in the HarmonicBalance namespace.

    source


    # HarmonicBalance.export_csv — Function.
    julia
    export_csv(filename, res, branch)

    Saves into filename a specified solution branch of the Result res.

    source


    + + + + \ No newline at end of file diff --git a/v0.10.7/manual/solving_harmonics.html b/v0.10.7/manual/solving_harmonics.html new file mode 100644 index 00000000..139dadbd --- /dev/null +++ b/v0.10.7/manual/solving_harmonics.html @@ -0,0 +1,71 @@ + + + + + + Solving harmonic equations | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Solving harmonic equations ​

    Once a differential equation of motion has been defined in DifferentialEquation and converted to a HarmonicEquation, we may use the homotopy continuation method (as implemented in HomotopyContinuation.jl) to find steady states. This means that, having called get_harmonic_equations, we need to set all time-derivatives to zero and parse the resulting algebraic equations into a Problem.

    Problem holds the steady-state equations, and (optionally) the symbolic Jacobian which is needed for stability / linear response calculations.

    Once defined, a Problem can be solved for a set of input parameters using get_steady_states to obtain Result.

    # HarmonicBalance.Problem — Type.
    julia
    mutable struct Problem

    Holds a set of algebraic equations describing the steady state of a system.

    Fields

    • variables::Vector{Num}: The harmonic variables to be solved for.

    • parameters::Vector{Num}: All symbols which are not the harmonic variables.

    • system::HomotopyContinuation.ModelKit.System: The input object for HomotopyContinuation.jl solver methods.

    • jacobian::Any: The Jacobian matrix (possibly symbolic). If false, the Jacobian is ignored (may be calculated implicitly after solving).

    • eom::HarmonicEquation: The HarmonicEquation object used to generate this Problem.

    Constructors

    julia
    Problem(eom::HarmonicEquation; Jacobian=true) # find and store the symbolic Jacobian
+Problem(eom::HarmonicEquation; Jacobian="implicit") # ignore the Jacobian for now, compute implicitly later
+Problem(eom::HarmonicEquation; Jacobian=J) # use J as the Jacobian (a function that takes a Dict)
+Problem(eom::HarmonicEquation; Jacobian=false) # ignore the Jacobian

    source


    # HarmonicBalance.get_steady_states — Function.
    julia
    get_steady_states(prob::Problem,
+                    swept_parameters::ParameterRange,
+                    fixed_parameters::ParameterList;
+                    method=:warmup,
+                    threading = Threads.nthreads() > 1,
+                    show_progress=true,
+                    sorting="nearest")

    Solves prob over the ranges specified by swept_parameters, keeping fixed_parameters constant. swept_parameters accepts pairs mapping symbolic variables to arrays or LinRange. fixed_parameters accepts pairs mapping symbolic variables to numbers.

    Keyword arguments

    • method: If :warmup (default), a problem similar to prob but with random complex parameters is first solved to find all non-singular paths. The subsequent tracking to find results for all swept_parameters is then much faster than the initial solving. If method=:total_degree, each parameter point is solved separately by tracking the maximum number of paths (employs a total degree homotopy).

    This takes far longer but can be more reliable.

    • threading: If true, multithreaded support is activated. The number of available threads is set by the environment variable JULIA_NUM_THREADS.

    • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".

    • show_progress: Indicate whether a progress bar should be displayed.

    Example: solving a simple harmonic oscillator mx¨+γx˙+ω02x=Fcos⁡(ωt) to obtain the response as a function of ω

    julia
    # having obtained a Problem object, let's find steady states
+julia> range = ParameterRange(ω => LinRange(0.8,1.2,100) ) # 100 parameter sets to solve
+julia> fixed = ParameterList(m => 1, γ => 0.01, F => 0.5, ω_0 => 1)
+julia> get_steady_states(problem, range, fixed)
+
+A steady state result for 100 parameter points
+
+    Solution branches:   1
+       of which real:    1
+       of which stable:  1
+
+    Classes: stable, physical, Hopf, binary_labels

    It is also possible to create multi-dimensional solutions plots.

    julia
    # The swept parameters take precedence over fixed -> use the same fixed
+julia> range = ParameterRange(ω => LinRange(0.8,1.2,100), F => LinRange(0.1,1.0,10) ) # 100x10 parameter sets
+
+# The swept parameters take precedence over fixed -> the F in fixed is now ignored
+julia> get_steady_states(problem, range, fixed)
+
+A steady state result for 1000 parameter points
+
+    Solution branches:   1
+       of which real:    1
+       of which stable:  1
+
+    Classes: stable, physical, Hopf, binary_labels

    source


    # HarmonicBalance.Result — Type.
    julia
    mutable struct Result

    Stores the steady states of a HarmonicEquation.

    Fields

    • solutions::Array{Vector{Vector{ComplexF64}}}: The variable values of steady-state solutions.

    • swept_parameters::OrderedCollections.OrderedDict{Num, Vector{Union{Float64, ComplexF64}}}: Values of all parameters for all solutions.

    • fixed_parameters::OrderedCollections.OrderedDict{Num, Float64}: The parameters fixed throughout the solutions.

    • problem::Problem: The Problem used to generate this.

    • classes::Dict{String, Array}: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method classify_solutions!).

    • jacobian::Function: The Jacobian with fixed_parameters already substituted. Accepts a dictionary specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function. If problem.jacobian was false, this holds a function that rearranges the equations to find J only after numerical values are inserted (preferable in cases where the symbolic J would be very large).

    • seed::Union{Nothing, UInt32}: Seed used for the solver

    source


    Classifying solutions ​

    The solutions in Result are accompanied by similarly-sized boolean arrays stored in the dictionary Result.classes. The classes can be used by the plotting functions to show/hide/label certain solutions.

    By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with classify_solutions!.

    # HarmonicBalance.classify_solutions! — Function.
    julia
    classify_solutions!(
+    res::Result,
+    func::Union{Function, String},
+    name::String;
+    physical
+) -> Any

    Creates a solution class in res using the inequality condition (parsed into Symbolics.jl input).

    The new class is labelled with name and stored under res.classes[name].

    By default, only physical (=real) solutions are classified, false is returned for the rest.

    Example

    julia
    # solve a previously-defined problem
+res = get_steady_states(problem, swept_parameters, fixed_parameters)
+
+# classify, store in result.classes["large_amplitude"]
+classify_solutions!(res, "sqrt(u1^2 + v1^2) > 1.0" , "large_amplitude")

    source


    Sorting solutions ​

    Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.

    For stable states, the branches describe a system's behaviour under adiabatic parameter changes.

    Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.

    The function sort_solutions goes over the the raw output of get_steady_states and sorts each entry such that neighboring solution sets minimize Euclidean distance.

    Currently, sort_solutions is compatible with 1D and 2D arrays of solution sets.

    # HarmonicBalance.sort_solutions — Function.
    julia
    sort_solutions(
+    solutions::Array;
+    sorting,
+    show_progress
+) -> Any

    Sorts solutions into branches according to the method sorting.

    solutions is an n-dimensional array of Vector{Vector}. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.

    Keyword arguments

    • sorting: the method used by sort_solutions to get continuous solutions branches. The current options are "hilbert" (1D sorting along a Hilbert curve), "nearest" (nearest-neighbor sorting) and "none".

    • show_progress: Indicate whether a progress bar should be displayed.

    source


    + + + + \ No newline at end of file diff --git a/v0.10.7/manual/time_dependent.html b/v0.10.7/manual/time_dependent.html new file mode 100644 index 00000000..9caa76a3 --- /dev/null +++ b/v0.10.7/manual/time_dependent.html @@ -0,0 +1,54 @@ + + + + + + Time evolution | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Time evolution ​

    Generally, solving the ODE of oscillatory systems in time requires numerically tracking the oscillations. This is a computationally expensive process; however, using the harmonic ansatz removes the oscillatory time-dependence. Simulating instead the harmonic variables of a HarmonicEquation is vastly more efficient - a steady state of the system appears as a fixed point in multidimensional space rather than an oscillatory function.

    The Extention TimeEvolution is used to interface HarmonicEquation with the solvers contained in OrdinaryDiffEq.jl. Time-dependent parameter sweeps are defined using the object ParameterSweep. To use the TimeEvolution extension, one must first load the OrdinaryDiffEq.jl package.

    # SciMLBase.ODEProblem — Method.
    julia
    ODEProblem(
+        eom::HarmonicEquation;
+        fixed_parameters,
+        x0::Vector,
+        sweep::ParameterSweep,
+        timespan::Tuple
+        )

    Creates an ODEProblem object used by OrdinaryDiffEq.jl from the equations in eom to simulate time-evolution within timespan. fixed_parameters must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If x0 is specified, it is used as an initial condition; otherwise the values from fixed_parameters are used.

    source


    # HarmonicBalance.ParameterSweep — Type.

    Represents a sweep of one or more parameters of a HarmonicEquation. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.

    Sweeps of different variables can be combined using +.

    Fields

    • functions::Dict{Num, Function}: Maps each swept parameter to a function.

    Examples

    julia
    # create a sweep of parameter a from 0 to 1 over time 0 -> 100
+julia> @variables a,b;
+julia> sweep = ParameterSweep(a => [0., 1.], (0, 100));
+julia> sweep[a](50)
+0.5
+julia> sweep[a](200)
+1.0
+
+# do the same, varying two parameters simultaneously
+julia> sweep = ParameterSweep([a => [0.,1.], b => [0., 1.]], (0,100))

    Successive sweeps can be combined,

    julia
    sweep1 = ParameterSweep(ω => [0.95, 1.0], (0, 2e4))
+sweep2 = ParameterSweep(λ => [0.05, 0.01], (2e4, 4e4))
+sweep = sweep1 + sweep2

    multiple parameters can be swept simultaneously,

    julia
    sweep = ParameterSweep([ω => [0.95;1.0], λ => [5e-2;1e-2]], (0, 2e4))

    and custom sweep functions may be used.

    julia
    ωfunc(t) = cos(t)
+sweep = ParameterSweep(ω => ωfunc)

    source


    Plotting ​

    # RecipesBase.plot — Method.
    julia
    plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

    Plot a function f of a time-dependent solution soln of harm_eq.

    As a function of time

    plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)

    f is parsed by Symbolics.jl

    parametric plots

    plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)

    Parametric plot of f[1] against f[2]

    Also callable as plot!

    source


    Miscellaneous ​

    Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.

    # HarmonicBalance.is_stable — Function.
    julia
    is_stable(
+    soln::OrderedCollections.OrderedDict{Num, ComplexF64},
+    eom::HarmonicEquation;
+    timespan,
+    tol,
+    perturb_initial
+)

    Numerically investigate the stability of a solution soln of eom within timespan. The initial condition is displaced by perturb_initial.

    Return true the solution evolves within tol of the initial value (interpreted as stable).

    source

    julia
    is_stable(
+    soln::OrderedCollections.OrderedDict{Num, ComplexF64},
+    res::Result;
+    kwargs...
+) -> Any

    Returns true if the solution soln of the Result res is stable. Stable solutions are real and have all Jacobian eigenvalues Re[λ] <= 0. im_tol : an absolute threshold to distinguish real/complex numbers. rel_tol: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)

    source


    + + + + \ No newline at end of file diff --git a/v0.10.7/siteinfo.js b/v0.10.7/siteinfo.js new file mode 100644 index 00000000..9013f6e9 --- /dev/null +++ b/v0.10.7/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.10.7"; diff --git a/v0.10.7/tutorials/classification.html b/v0.10.7/tutorials/classification.html new file mode 100644 index 00000000..f53ac23c --- /dev/null +++ b/v0.10.7/tutorials/classification.html @@ -0,0 +1,70 @@ + + + + + + Classifying solutions | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Classifying solutions ​

    Given that you obtained some steady states for a parameter sweep of a specific model it can be useful to classify these solution. Let us consider a simple pametric oscillator

    julia
    using HarmonicBalance
+
+@variables ω₀ γ λ α ω t x(t)
+
+natural_equation = d(d(x, t), t) + γ * d(x, t) + (ω₀^2 - λ * cos(2 * ω * t)) * x + α * x^3
+diff_eq = DifferentialEquation(natural_equation, x)
+
+add_harmonic!(diff_eq, x, ω);
+
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω₀, λ
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+-(1//2)*u1(T)*λ + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω₀^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ 0
+
+Differential(T)(v1(T))*γ + (1//2)*v1(T)*λ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω₀^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0

    We performe a 2d sweep in the driving frequency ω and driving strength λ:

    julia
    fixed = (ω₀ => 1.0, γ => 0.002, α => 1.0)
+varied = (ω => range(0.99, 1.01, 100), λ => range(1e-6, 0.03, 100))
+
+result_2D = get_steady_states(harmonic_eq, varied, fixed, threading=true)
    A steady state result for 10000 parameter points
+
+Solution branches:   5
+   of which real:    5
+   of which stable:  5
+
+Classes: stable, physical, Hopf, binary_labels

    By default the steady states of the system are classified by four different catogaries:

    • physical: Solutions that are physical, i.e., all variables are purely real.

    • stable: Solutions that are stable, i.e., all eigenvalues of the Jacobian have negative real parts.

    • Hopf: Solutions that are physical and have exactly two Jacobian eigenvalues with positive real parts, which are complex conjugates of each other. The class can help to identify regions where a limit cycle is present due to a Hopf bifurcation. See also the tutorial on limit cycles.

    • binary_labels: each region in the parameter sweep receives an identifier based on its permutation of stable branches. This allows to distinguish between different phases, which may have the same number of stable solutions.

    We can plot the number of stable solutions, giving the phase diagram

    julia
    plot_phase_diagram(result_2D, class="stable")

    If we plot the a cut at λ=0.01, we see that in the blue region only one stable solution exists with zero amplitude:

    julia
    plot(result_2D, y="√(u1^2+v1^2)", cut=λ => 0.01, class="stable") |> display
    julia
    get_single_solution(result_2D; branch=1, index=(1, 1))
    OrderedCollections.OrderedDict{Num, ComplexF64} with 7 entries:
+  u1 => -3.35208e-249-5.36333e-248im
+  v1 => -7.59806e-248+1.45257e-248im
+  ω  => 0.99+0.0im
+  λ  => 1.0e-6+0.0im
+  ω₀ => 1.0+0.0im
+  γ  => 0.002+0.0im
+  α  => 1.0+0.0im

    This solution becomes stable again outside the green lobe. Also called Mathieu lobe. Indeed, we can classify the zero amplitude solution by adding an extra catagory as a class:

    julia
    classify_solutions!(result_2D, "sqrt(u1^2 + v1^2) < 0.001", "zero")
+result_2D
    A steady state result for 10000 parameter points
+
+Solution branches:   5
+   of which real:    5
+   of which stable:  5
+
+Classes: zero, stable, physical, Hopf, binary_labels

    We can visualize the zero amplitude solution:

    julia
    plot_phase_diagram(result_2D, class=["zero", "stable"])

    This shows that inside the Mathieu lobe the zero amplitude solution becomes unstable due to the parametric drive being resonant with the oscillator.

    We can also visualize the equi-amplitude curves of the solutions:

    julia
    classify_solutions!(result_2D, "sqrt(u1^2 + v1^2) > 0.12", "large amplitude")
+plot_phase_diagram(result_2D, class=["large amplitude", "stable"])

    + + + + \ No newline at end of file diff --git a/v0.10.7/tutorials/index.html b/v0.10.7/tutorials/index.html new file mode 100644 index 00000000..fe58d909 --- /dev/null +++ b/v0.10.7/tutorials/index.html @@ -0,0 +1,27 @@ + + + + + + Tutorials | HarmonicBalance.jl + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/v0.10.7/tutorials/limit_cycles.html b/v0.10.7/tutorials/limit_cycles.html new file mode 100644 index 00000000..e2f7b217 --- /dev/null +++ b/v0.10.7/tutorials/limit_cycles.html @@ -0,0 +1,116 @@ + + + + + + Limit cycles | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Limit cycles ​

    In contrast to the previous tutorials, limit cycle problems feature harmonic(s) whose numerical value is not imposed externally. We shall construct our HarmonicEquation as usual, but identify this harmonic as an extra variable, rather than a fixed parameter.

    Non-driven system - the van der Pol oscillator ​

    Here we solve the equation of motion of the van der Pol oscillator. This is a single-variable second-order ODE with continuous time-translation symmetry (i.e., no 'clock' imposing a frequency and/or phase), which displays periodic solutions known as relaxation oscillations. For more detail, refer also to arXiv:2308.06092.

    julia
    using HarmonicBalance
+@variables ω_lc, t, ω0, x(t), μ
+diff_eq = DifferentialEquation(d(d(x,t),t) - μ*(1-x^2) * d(x,t) + x, x)
    System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ;   
+
+x(t) + Differential(t)(Differential(t)(x(t))) - (1 - (x(t)^2))*Differential(t)(x(t))*μ ~ 0

    Choosing to expand the motion of x(t) using ωlc, 3ωlc and 5ωlc, we define

    julia
    foreach(1:2:5) do i
+  add_harmonic!(diff_eq, x, i*ω_lc)
+end;

    and obtain 6 harmonic equations,

    julia
    harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 6 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T), u3(T), v3(T)
+Parameters: ω_lc, μ
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ω_lct) + v1(T)*sin(ω_lct) + u2(T)*cos(3ω_lct) + v2(T)*sin(3ω_lct) + u3(T)*cos(5ω_lct) + v3(T)*sin(5ω_lct)
+
+Harmonic equations:
+
+u1(T) + (2//1)*Differential(T)(v1(T))*ω_lc - Differential(T)(u1(T))*μ - u1(T)*(ω_lc^2) - v1(T)*μ*ω_lc + (1//4)*(u1(T)^2)*Differential(T)(u2(T))*μ + (3//4)*(u1(T)^2)*Differential(T)(u1(T))*μ + u1(T)*Differential(T)(u3(T))*u3(T)*μ + (1//2)*u1(T)*Differential(T)(u3(T))*u2(T)*μ + u1(T)*Differential(T)(v3(T))*v3(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*v2(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*Differential(T)(v1(T))*v2(T)*μ + (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*μ + u1(T)*v2(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*Differential(T)(u2(T))*u3(T)*μ + u1(T)*Differential(T)(u2(T))*u2(T)*μ + (1//2)*u1(T)*Differential(T)(u1(T))*u2(T)*μ - (1//4)*Differential(T)(u3(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(u3(T))*v2(T)*v1(T)*μ + (1//4)*Differential(T)(u3(T))*(u2(T)^2)*μ + (1//2)*Differential(T)(v3(T))*v2(T)*u2(T)*μ + (1//2)*Differential(T)(v3(T))*v1(T)*u2(T)*μ + (1//2)*(v3(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v3(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*v3(T)*v2(T)*Differential(T)(u2(T))*μ + (1//2)*v3(T)*v2(T)*Differential(T)(u1(T))*μ + (1//2)*v3(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*v3(T)*u2(T)*Differential(T)(v2(T))*μ - (1//2)*Differential(T)(v1(T))*v2(T)*u3(T)*μ - (1//2)*Differential(T)(v1(T))*v1(T)*u2(T)*μ + (1//2)*(v2(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v2(T)*v1(T)*Differential(T)(u1(T))*μ - (1//2)*v2(T)*u3(T)*Differential(T)(v2(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(u2(T))*μ + (1//4)*(v1(T)^2)*Differential(T)(u1(T))*μ - (1//2)*v1(T)*u3(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u2(T))*u3(T)*u2(T)*μ + (1//2)*(u3(T)^2)*Differential(T)(u1(T))*μ + (1//2)*u3(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*Differential(T)(u1(T))*(u2(T)^2)*μ + (1//4)*(u1(T)^2)*v2(T)*μ*ω_lc + (1//4)*(u1(T)^2)*v1(T)*μ*ω_lc + (1//2)*u1(T)*v3(T)*u2(T)*μ*ω_lc - (1//2)*u1(T)*v2(T)*u3(T)*μ*ω_lc - (1//2)*u1(T)*v1(T)*u2(T)*μ*ω_lc + (1//2)*(v3(T)^2)*v1(T)*μ*ω_lc + (1//4)*v3(T)*(v2(T)^2)*μ*ω_lc - (1//2)*v3(T)*v2(T)*v1(T)*μ*ω_lc - (1//4)*v3(T)*(u2(T)^2)*μ*ω_lc + (1//2)*(v2(T)^2)*v1(T)*μ*ω_lc - (1//4)*v2(T)*(v1(T)^2)*μ*ω_lc + (1//2)*v2(T)*u3(T)*u2(T)*μ*ω_lc + (1//4)*(v1(T)^3)*μ*ω_lc + (1//2)*v1(T)*(u3(T)^2)*μ*ω_lc - (1//2)*v1(T)*u3(T)*u2(T)*μ*ω_lc + (1//2)*v1(T)*(u2(T)^2)*μ*ω_lc ~ 0
+
+v1(T) - Differential(T)(v1(T))*μ - (2//1)*Differential(T)(u1(T))*ω_lc + u1(T)*μ*ω_lc - v1(T)*(ω_lc^2) + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*Differential(T)(u3(T))*v2(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*u2(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(u2(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*u1(T)*v2(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*u3(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u3(T))*v2(T)*u2(T)*μ + Differential(T)(u3(T))*v1(T)*u3(T)*μ - (1//2)*Differential(T)(u3(T))*v1(T)*u2(T)*μ + Differential(T)(v3(T))*v3(T)*v1(T)*μ + (1//4)*Differential(T)(v3(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(v3(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v3(T))*(u2(T)^2)*μ + (1//2)*(v3(T)^2)*Differential(T)(v1(T))*μ - (1//2)*v3(T)*Differential(T)(v1(T))*v2(T)*μ + (1//2)*v3(T)*v2(T)*Differential(T)(v2(T))*μ - (1//2)*v3(T)*v1(T)*Differential(T)(v2(T))*μ - (1//2)*v3(T)*Differential(T)(u2(T))*u2(T)*μ + (1//2)*v3(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*Differential(T)(v1(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(v1(T))*v2(T)*v1(T)*μ + (3//4)*Differential(T)(v1(T))*(v1(T)^2)*μ + (1//2)*Differential(T)(v1(T))*(u3(T)^2)*μ - (1//2)*Differential(T)(v1(T))*u3(T)*u2(T)*μ + (1//2)*Differential(T)(v1(T))*(u2(T)^2)*μ + v2(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*v2(T)*Differential(T)(u2(T))*u3(T)*μ - (1//2)*v2(T)*u3(T)*Differential(T)(u1(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*v1(T)*Differential(T)(u2(T))*u3(T)*μ + v1(T)*Differential(T)(u2(T))*u2(T)*μ - (1//2)*v1(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*u3(T)*u2(T)*Differential(T)(v2(T))*μ - (1//4)*(u1(T)^3)*μ*ω_lc - (1//4)*(u1(T)^2)*u2(T)*μ*ω_lc - (1//2)*u1(T)*(v3(T)^2)*μ*ω_lc - (1//2)*u1(T)*v3(T)*v2(T)*μ*ω_lc - (1//2)*u1(T)*(v2(T)^2)*μ*ω_lc - (1//2)*u1(T)*v2(T)*v1(T)*μ*ω_lc - (1//4)*u1(T)*(v1(T)^2)*μ*ω_lc - (1//2)*u1(T)*(u3(T)^2)*μ*ω_lc - (1//2)*u1(T)*u3(T)*u2(T)*μ*ω_lc - (1//2)*u1(T)*(u2(T)^2)*μ*ω_lc - (1//2)*v3(T)*v2(T)*u2(T)*μ*ω_lc - (1//2)*v3(T)*v1(T)*u2(T)*μ*ω_lc + (1//4)*(v2(T)^2)*u3(T)*μ*ω_lc + (1//2)*v2(T)*v1(T)*u3(T)*μ*ω_lc + (1//4)*(v1(T)^2)*u2(T)*μ*ω_lc - (1//4)*u3(T)*(u2(T)^2)*μ*ω_lc ~ 0
+
+u2(T) - Differential(T)(u2(T))*μ + (6//1)*Differential(T)(v2(T))*ω_lc - (3//1)*v2(T)*μ*ω_lc - (9//1)*u2(T)*(ω_lc^2) + (1//4)*(u1(T)^2)*Differential(T)(u3(T))*μ + (1//2)*(u1(T)^2)*Differential(T)(u2(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*Differential(T)(u3(T))*u2(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*v2(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*v1(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(v1(T))*μ + (1//2)*u1(T)*v3(T)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*μ + (1//2)*u1(T)*Differential(T)(u2(T))*u3(T)*μ + (1//2)*u1(T)*u3(T)*Differential(T)(u1(T))*μ + u1(T)*Differential(T)(u1(T))*u2(T)*μ + (1//2)*Differential(T)(u3(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(u3(T))*(v1(T)^2)*μ + Differential(T)(u3(T))*u3(T)*u2(T)*μ + Differential(T)(v3(T))*v3(T)*u2(T)*μ - (1//2)*Differential(T)(v3(T))*v1(T)*u2(T)*μ + (1//2)*(v3(T)^2)*Differential(T)(u2(T))*μ - (1//2)*v3(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*v3(T)*v2(T)*Differential(T)(u1(T))*μ - (1//2)*v3(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*v3(T)*v1(T)*Differential(T)(u1(T))*μ + (1//2)*Differential(T)(v1(T))*v2(T)*u3(T)*μ - (1//2)*Differential(T)(v1(T))*v1(T)*u3(T)*μ + Differential(T)(v1(T))*v1(T)*u2(T)*μ + (1//4)*(v2(T)^2)*Differential(T)(u2(T))*μ + (1//2)*v2(T)*u2(T)*Differential(T)(v2(T))*μ + (1//2)*(v1(T)^2)*Differential(T)(u2(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v1(T)*u3(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u2(T))*(u3(T)^2)*μ + (3//4)*Differential(T)(u2(T))*(u2(T)^2)*μ + (1//2)*u3(T)*Differential(T)(u1(T))*u2(T)*μ + (3//4)*(u1(T)^2)*v3(T)*μ*ω_lc + (3//2)*(u1(T)^2)*v2(T)*μ*ω_lc + (3//4)*(u1(T)^2)*v1(T)*μ*ω_lc + (3//2)*u1(T)*v3(T)*u2(T)*μ*ω_lc - (3//2)*u1(T)*v2(T)*u3(T)*μ*ω_lc - (3//2)*u1(T)*v1(T)*u3(T)*μ*ω_lc + (3//2)*(v3(T)^2)*v2(T)*μ*ω_lc + (3//2)*v3(T)*v2(T)*v1(T)*μ*ω_lc - (3//4)*v3(T)*(v1(T)^2)*μ*ω_lc + (3//4)*(v2(T)^3)*μ*ω_lc + (3//2)*v2(T)*(v1(T)^2)*μ*ω_lc + (3//2)*v2(T)*(u3(T)^2)*μ*ω_lc + (3//4)*v2(T)*(u2(T)^2)*μ*ω_lc - (1//4)*(v1(T)^3)*μ*ω_lc + (3//2)*v1(T)*u3(T)*u2(T)*μ*ω_lc ~ 0
+
+v2(T) - (6//1)*Differential(T)(u2(T))*ω_lc - Differential(T)(v2(T))*μ - (9//1)*v2(T)*(ω_lc^2) + (3//1)*u2(T)*μ*ω_lc + (1//4)*(u1(T)^2)*Differential(T)(v3(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*μ + (1//2)*(u1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*Differential(T)(u3(T))*v2(T)*μ - (1//2)*u1(T)*Differential(T)(u3(T))*v1(T)*μ + (1//2)*u1(T)*Differential(T)(v3(T))*u2(T)*μ + (1//2)*u1(T)*v3(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*v3(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*u3(T)*μ + u1(T)*v2(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*μ - (1//2)*u1(T)*u3(T)*Differential(T)(v2(T))*μ + Differential(T)(u3(T))*v2(T)*u3(T)*μ + (1//2)*Differential(T)(u3(T))*v1(T)*u2(T)*μ + Differential(T)(v3(T))*v3(T)*v2(T)*μ + (1//2)*Differential(T)(v3(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v3(T))*(v1(T)^2)*μ + (1//2)*(v3(T)^2)*Differential(T)(v2(T))*μ + (1//2)*v3(T)*Differential(T)(v1(T))*v2(T)*μ - (1//2)*v3(T)*Differential(T)(v1(T))*v1(T)*μ + (1//2)*v3(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*v3(T)*Differential(T)(u1(T))*u2(T)*μ + Differential(T)(v1(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v1(T))*(v1(T)^2)*μ + (1//2)*Differential(T)(v1(T))*u3(T)*u2(T)*μ + (3//4)*(v2(T)^2)*Differential(T)(v2(T))*μ + (1//2)*v2(T)*Differential(T)(u2(T))*u2(T)*μ - (1//2)*v2(T)*u3(T)*Differential(T)(u1(T))*μ + (1//2)*(v1(T)^2)*Differential(T)(v2(T))*μ + (1//2)*v1(T)*Differential(T)(u2(T))*u3(T)*μ - (1//2)*v1(T)*u3(T)*Differential(T)(u1(T))*μ + (1//2)*(u3(T)^2)*Differential(T)(v2(T))*μ + (1//4)*(u2(T)^2)*Differential(T)(v2(T))*μ - (1//4)*(u1(T)^3)*μ*ω_lc - (3//4)*(u1(T)^2)*u3(T)*μ*ω_lc - (3//2)*(u1(T)^2)*u2(T)*μ*ω_lc - (3//2)*u1(T)*v3(T)*v2(T)*μ*ω_lc - (3//2)*u1(T)*v3(T)*v1(T)*μ*ω_lc + (3//4)*u1(T)*(v1(T)^2)*μ*ω_lc - (3//2)*u1(T)*u3(T)*u2(T)*μ*ω_lc - (3//2)*(v3(T)^2)*u2(T)*μ*ω_lc + (3//2)*v3(T)*v1(T)*u2(T)*μ*ω_lc - (3//4)*(v2(T)^2)*u2(T)*μ*ω_lc - (3//2)*v2(T)*v1(T)*u3(T)*μ*ω_lc + (3//4)*(v1(T)^2)*u3(T)*μ*ω_lc - (3//2)*(v1(T)^2)*u2(T)*μ*ω_lc - (3//2)*(u3(T)^2)*u2(T)*μ*ω_lc - (3//4)*(u2(T)^3)*μ*ω_lc ~ 0
+
+u3(T) - Differential(T)(u3(T))*μ + (10//1)*Differential(T)(v3(T))*ω_lc - (5//1)*v3(T)*μ*ω_lc - (25//1)*u3(T)*(ω_lc^2) + (1//2)*(u1(T)^2)*Differential(T)(u3(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(u2(T))*μ - (1//2)*u1(T)*Differential(T)(v1(T))*v2(T)*μ - (1//2)*u1(T)*v2(T)*Differential(T)(v2(T))*μ - (1//2)*u1(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*u1(T)*Differential(T)(u2(T))*u2(T)*μ + u1(T)*u3(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*Differential(T)(u1(T))*u2(T)*μ + (1//4)*Differential(T)(u3(T))*(v3(T)^2)*μ + (1//2)*Differential(T)(u3(T))*(v2(T)^2)*μ + (1//2)*Differential(T)(u3(T))*(v1(T)^2)*μ + (3//4)*Differential(T)(u3(T))*(u3(T)^2)*μ + (1//2)*Differential(T)(u3(T))*(u2(T)^2)*μ + (1//2)*Differential(T)(v3(T))*v3(T)*u3(T)*μ + (1//2)*Differential(T)(v1(T))*v2(T)*u2(T)*μ + Differential(T)(v1(T))*v1(T)*u3(T)*μ - (1//2)*Differential(T)(v1(T))*v1(T)*u2(T)*μ - (1//4)*(v2(T)^2)*Differential(T)(u1(T))*μ + (1//2)*v2(T)*v1(T)*Differential(T)(u2(T))*μ - (1//2)*v2(T)*v1(T)*Differential(T)(u1(T))*μ + v2(T)*u3(T)*Differential(T)(v2(T))*μ - (1//4)*(v1(T)^2)*Differential(T)(u2(T))*μ + (1//2)*v1(T)*u2(T)*Differential(T)(v2(T))*μ + Differential(T)(u2(T))*u3(T)*u2(T)*μ + (1//4)*Differential(T)(u1(T))*(u2(T)^2)*μ + (5//2)*(u1(T)^2)*v3(T)*μ*ω_lc + (5//4)*(u1(T)^2)*v2(T)*μ*ω_lc + (5//2)*u1(T)*v2(T)*u2(T)*μ*ω_lc + (5//2)*u1(T)*v1(T)*u2(T)*μ*ω_lc + (5//4)*(v3(T)^3)*μ*ω_lc + (5//2)*v3(T)*(v2(T)^2)*μ*ω_lc + (5//2)*v3(T)*(v1(T)^2)*μ*ω_lc + (5//4)*v3(T)*(u3(T)^2)*μ*ω_lc + (5//2)*v3(T)*(u2(T)^2)*μ*ω_lc + (5//4)*(v2(T)^2)*v1(T)*μ*ω_lc - (5//4)*v2(T)*(v1(T)^2)*μ*ω_lc - (5//4)*v1(T)*(u2(T)^2)*μ*ω_lc ~ 0
+
+v3(T) - (10//1)*Differential(T)(u3(T))*ω_lc - Differential(T)(v3(T))*μ - (25//1)*v3(T)*(ω_lc^2) + (5//1)*u3(T)*μ*ω_lc + (1//2)*(u1(T)^2)*Differential(T)(v3(T))*μ + (1//4)*(u1(T)^2)*Differential(T)(v2(T))*μ + u1(T)*v3(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*Differential(T)(v1(T))*u2(T)*μ + (1//2)*u1(T)*v2(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*v2(T)*Differential(T)(u1(T))*μ + (1//2)*u1(T)*v1(T)*Differential(T)(u2(T))*μ + (1//2)*u1(T)*u2(T)*Differential(T)(v2(T))*μ + (1//2)*Differential(T)(u3(T))*v3(T)*u3(T)*μ + (3//4)*Differential(T)(v3(T))*(v3(T)^2)*μ + (1//2)*Differential(T)(v3(T))*(v2(T)^2)*μ + (1//2)*Differential(T)(v3(T))*(v1(T)^2)*μ + (1//4)*Differential(T)(v3(T))*(u3(T)^2)*μ + (1//2)*Differential(T)(v3(T))*(u2(T)^2)*μ + v3(T)*Differential(T)(v1(T))*v1(T)*μ + v3(T)*v2(T)*Differential(T)(v2(T))*μ + v3(T)*Differential(T)(u2(T))*u2(T)*μ + (1//4)*Differential(T)(v1(T))*(v2(T)^2)*μ - (1//2)*Differential(T)(v1(T))*v2(T)*v1(T)*μ - (1//4)*Differential(T)(v1(T))*(u2(T)^2)*μ + (1//2)*v2(T)*v1(T)*Differential(T)(v2(T))*μ + (1//2)*v2(T)*Differential(T)(u1(T))*u2(T)*μ - (1//4)*(v1(T)^2)*Differential(T)(v2(T))*μ - (1//2)*v1(T)*Differential(T)(u2(T))*u2(T)*μ + (1//2)*v1(T)*Differential(T)(u1(T))*u2(T)*μ - (5//2)*(u1(T)^2)*u3(T)*μ*ω_lc - (5//4)*(u1(T)^2)*u2(T)*μ*ω_lc + (5//4)*u1(T)*(v2(T)^2)*μ*ω_lc + (5//2)*u1(T)*v2(T)*v1(T)*μ*ω_lc - (5//4)*u1(T)*(u2(T)^2)*μ*ω_lc - (5//4)*(v3(T)^2)*u3(T)*μ*ω_lc - (5//2)*(v2(T)^2)*u3(T)*μ*ω_lc - (5//2)*v2(T)*v1(T)*u2(T)*μ*ω_lc - (5//2)*(v1(T)^2)*u3(T)*μ*ω_lc + (5//4)*(v1(T)^2)*u2(T)*μ*ω_lc - (5//4)*(u3(T)^3)*μ*ω_lc - (5//2)*u3(T)*(u2(T)^2)*μ*ω_lc ~ 0

    So far, ωlc appears as any other harmonic. However, it is not fixed by any external drive or 'clock', instead, it emerges out of a Hopf instability in the system. We can verify that fixing ω_lc and calling get_steady_states.

    julia
    get_steady_states(harmonic_eq, μ => 1:0.1:5, ω_lc => 1.2)

    gives a single solution with zero amplitude.

    Taking instead ωlc as a variable to be solved for results in a phase freedom, implying an infinite number of solutions. To perform the gauge-fixing procedure, we call get_limit_cycles, marking the limit cycle harmonic as a keyword argument,

    julia
    result = get_limit_cycles(harmonic_eq, μ => 1:0.1:5, (), ω_lc)
    A steady state result for 41 parameter points
+
+Solution branches:   100
+   of which real:    4
+   of which stable:  4
+
+Classes: unique_cycle, stable, physical, Hopf, binary_labels

    The results show a fourfold degeneracy of solutions:

    julia
    plot(result, y="ω_lc")

    The automatically created solution class unique_cycle filters the degeneracy out:

    julia
    plot(result, y="ω_lc", class="unique_cycle")

    Driven system - coupled Duffings ​

    So far, we have largely focused on finding and analysing steady states, i.e., fixed points of the harmonic equations, which satisfy

    du(T)dT=F¯(u)=0.

    Fixed points are however merely a subset of possible solutions of Eq. \eqref{eq:harmeqfull} – strictly speaking, solutions where u(T) remains time-dependent are allowed. These are quite unusual, since F¯(u) is by construction time-independent and Eq. \eqref{eq:harmeqfull} thus possesses continuous time-translation symmetry. The appearance of explicitly time-dependent solutions then consitutes spontaneous time-translation symmetry breaking.

    Such solutions, known as limit cycles, typically appear as closed periodic trajectories of the harmonic variables u(T). The simplest way to numerically characterise them is a time-dependent simulation, using a steady-state diagram as a guide.

    Here we reconstruct the results of Zambon et al., Phys Rev. A 102, 023526 (2020), where limit cycles are shown to appear in a system of two coupled nonlinear oscillators. In this problem, two oscillators x1 and x2, have (the same) damping and Kerr nonlinearity and are linearly coupled,

    x¨1+γx˙1+ω02x1+αx13+2J(x1−x2)=F0cos⁡(ωt)x¨2+γx˙2+ω02x2+αx23+2J(x2−x1)=ηF0cos⁡(ωt)
    julia
    using HarmonicBalance
+@variables γ F α ω0 F0 η ω J t x(t) y(t);
+eqs = [d(x,t,2) + γ*d(x,t) + ω0^2*x + α*x^3+ 2*J*ω0*(x-y) - F0*cos(ω*t),
+       d(y,t,2) + γ * d(y,t) + ω0^2 * y + α*y^3 + 2*J*ω0*(y-x) - η*F0*cos(ω*t)]
+diff_eq = DifferentialEquation(eqs, [x,y])
    System of 2 differential equations
+Variables:       x(t), y(t)
+Harmonic ansatz: x(t) => ;   y(t) => ;   
+
+Differential(t)(Differential(t)(x(t))) - F0*cos(t*ω) + Differential(t)(x(t))*γ + 2J*(x(t) - y(t))*ω0 + x(t)*(ω0^2) + (x(t)^3)*α ~ 0
+Differential(t)(Differential(t)(y(t))) + Differential(t)(y(t))*γ - F0*cos(t*ω)*η + 2J*(-x(t) + y(t))*ω0 + y(t)*(ω0^2) + (y(t)^3)*α ~ 0

    The analysis of Zambon et al. uses a frame rotating at the pump frequency ω to describe both oscillators. For us, this means we expand both modes using ω to obtain the harmonic equations.

    julia
    add_harmonic!(diff_eq, x, ω)
+add_harmonic!(diff_eq, y, ω)
+
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 4 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T)
+Parameters: ω, ω0, J, α, γ, F0, η
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+y(t) = u2(T)*cos(ωt) + v2(T)*sin(ωt)
+
+Harmonic equations:
+
+-F0 + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ + (2//1)*J*u1(T)*ω0 - (2//1)*J*u2(T)*ω0 - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ 0
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - (2//1)*J*v2(T)*ω0 + (2//1)*J*v1(T)*ω0 - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0
+
+-F0*η + Differential(T)(u2(T))*γ + (2//1)*Differential(T)(v2(T))*ω - (2//1)*J*u1(T)*ω0 + (2//1)*J*u2(T)*ω0 + v2(T)*γ*ω - u2(T)*(ω^2) + u2(T)*(ω0^2) + (3//4)*(v2(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0
+
+-(2//1)*Differential(T)(u2(T))*ω + Differential(T)(v2(T))*γ + (2//1)*J*v2(T)*ω0 - (2//1)*J*v1(T)*ω0 - v2(T)*(ω^2) + v2(T)*(ω0^2) - u2(T)*γ*ω + (3//4)*(v2(T)^3)*α + (3//4)*v2(T)*(u2(T)^2)*α ~ 0

    Solving for a range of drive amplitudes F0,

    julia
    fixed = (
+    ω0 => 1.4504859, # natural frequency of separate modes (in paper's notation, ħω0 - J)
+    γ => 27.4e-6,    # damping
+    J => 154.1e-6,   # coupling term
+    α => 3.867e-7,   # Kerr nonlinearity
+    ω => 1.4507941,  # pump frequency, resonant with antisymmetric mode (in paper, ħω0 + J)
+    η => -0.08,      # pumping leaking to site 2  (F2 = ηF1)
+    F0 => 0.002       # pump amplitude (overriden in sweeps)
+)
+varied = F0 => range(0.002, 0.03, 50)
+
+result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 50 parameter points
+
+Solution branches:   9
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels

    Let us first see the steady states.

    julia
    p1 = plot(result, "u1^2 + v1^2", legend=false)
+p2 = plot(result, "u2^2 + v2^2")
+plot(p1, p2)

    According to Zambon et al., a limit cycle solution exists around F0≅0.011, which can be accessed by a jump from branch 1 in an upwards sweep of F0. Since a limit cycle is not a steady state of our harmonic equations, it does not appear in the diagram. We do however see that branch 1 ceases to be stable around F0≅0.010, meaning a jump should occur.

    Let us try and simulate the limit cycle. We could in principle run a time-dependent simulation with a fixed value of F0, but this would require a suitable initial condition. Instead, we will sweep F0 upwards from a low starting value. To observe the dynamics just after the jump has occurred, we follow the sweep by a time interval where the system evolves under fixed parameters.

    julia
    using OrdinaryDiffEq
+initial_state = result[1][1]
+
+T = 2e6
+sweep = ParameterSweep(F0 => (0.002, 0.011), (0,T))
+
+# start from initial_state, use sweep, total time is 2*T
+time_problem = ODEProblem(harmonic_eq, initial_state, sweep=sweep, timespan=(0,2*T))
+time_evo = solve(time_problem, saveat=100);

    Inspecting the amplitude as a function of time,

    julia
    plot(time_evo, "sqrt(u1^2 + v1^2)", harmonic_eq)

    we see that initially the sweep is adiabatic as it proceeds along the steady-state branch 1. At around T=2e6, an instability occurs and u1(T) starts to rapidly oscillate. At that point, the sweep is stopped. Under free time evolution, the system then settles into a limit-cycle solution where the coordinates move along closed trajectories.

    By plotting the u and v variables against each other, we observe the limit cycle shapes in phase space,

    julia
    p1 = plot(time_evo, ["u1", "v1"], harmonic_eq)
+p2 = plot(time_evo, ["u2", "v2"], harmonic_eq)
+plot(p1, p2)

    + + + + \ No newline at end of file diff --git a/v0.10.7/tutorials/linear_response.html b/v0.10.7/tutorials/linear_response.html new file mode 100644 index 00000000..20450e97 --- /dev/null +++ b/v0.10.7/tutorials/linear_response.html @@ -0,0 +1,76 @@ + + + + + + Linear response | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

    Appearance

    Linear response ​

    In HarmonicBalance.jl, the stability and linear response are treated using the LinearResponse module.

    Here we calculate the white noise response of a simple nonlinear system. A set of reference results may be found in Huber et al. in Phys. Rev. X 10, 021066 (2020). We start by defining the Duffing oscillator

    julia
    using HarmonicBalance, Plots
+using Plots.Measures: mm
+@variables α, ω, ω0, F, γ, t, x(t); # declare constant variables and a function x(t)
+
+# define ODE
+diff_eq = DifferentialEquation(d(x,t,2) + ω0*x + α*x^3 + γ*d(x,t) ~ F*cos(ω*t), x)
+
+# specify the ansatz x = u(T) cos(ω*t) + v(T) sin(ω*t)
+add_harmonic!(diff_eq, x, ω)
+
+# implement ansatz to get harmonic equations
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω0, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+u1(T)*ω0 + (2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ F
+
+Differential(T)(v1(T))*γ + v1(T)*ω0 - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0

    Linear regime ​

    When driven weakly, the Duffing resonator behaves quasi-linearly, i.e, its response to noise is independent of the applied drive. We see that for weak driving, F=10−4, the amplitude is a Lorentzian.

    julia
    fixed = (α => 1, ω0 => 1.0, γ => 0.005, F => 0.0001)   # fixed parameters
+varied = ω => range(0.95, 1.05, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
+
+plot(result, "sqrt(u1^2 + v1^2)")

    To find the fluctuation on the top of the steady state one often employs a Bogoliubov-de Gennes analyses. Here, we compute the eigenvalues λk of the Jacobian matrix at the steady state. The imaginary part of the eigenvalues gives characteristic frequencies of the "quasi-particle excitations". The real part gives the lifetime of these excitations.

    One can plot the eigenvalues as follows

    julia
    plot(
+    plot_eigenvalues(result, branch=1),
+    plot_eigenvalues(result, branch=1, type=:real, ylims=(-0.003, 0)),
+)

    We find a single pair of complex conjugate eigenvalues linearly changing with the driving frequency. Both real parts are negative, indicating stability.

    As discussed in background section on linear response, the excitation manifest itself as a lorentenzian peak in a power spectral density (PSD) measurement. The PSD can be plotted using plot_linear_response:

    julia
    plot_linear_response(result, x, Ω_range=range(0.95, 1.05, 300), branch=1, logscale=true)

    The response has a peak at ω0, irrespective of the driving frequency ω. Indeed, the eigenvalues shown before where plotted in the rotating frame at the frequency of the drive ω. Hence, the imaginary part of eigenvalues shows the frequency (energy) needed to excite the system at it natural frequency (The frequency its want to be excited at.)

    Note the slight "bending" of the noise peak with ω - this is given by the failure of the first-order calculation to capture response far-detuned from the drive frequency.

    Nonlinear regime ​

    For strong driving, matters get more complicated. Let us now use a drive F=2∗10−3 :

    julia
    fixed = (α => 1, ω0 => 1.0, γ => 0.005, F => 0.002)   # fixed parameters
+varied = ω => range(0.95, 1.05, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
+
+plot(result, x="ω", y="sqrt(u1^2 + v1^2)");

    The amplitude is the well-known Duffing curve. Let's look at the eigenvalues of the two stable branches, 1 and 2.

    julia
    plot(
+    plot_eigenvalues(result, branch=1),
+    plot_eigenvalues(result, branch=1, type=:real, ylims=(-0.003, 0)),
+    plot_eigenvalues(result, branch=2),
+    plot_eigenvalues(result, branch=2, type=:real, ylims=(-0.003, 0)),
+)

    Again every branch gives a single pair of complex conjugate eigenvalues. However, for branch 1, the characteristic frequencies due not change linearly with the driving frequency around ω=ω0. This is a sign of steady state becoming nonlinear at large amplitudes.

    The same can be seen in the PSD:

    julia
    plot(
+  plot_linear_response(result, x, branch=1, Ω_range=range(0.95,1.1,300), logscale=true),
+  plot_linear_response(result, x, branch=2, Ω_range=range(0.9,1.1,300), logscale=true),
+    size=(600, 250), margin=3mm
+)

    In branch 1 the linear response to white noise shows more than one peak. This is a distinctly nonlinear phenomenon, indicitive if the squeezing of the steady state. Branch 2 is again quasi-linear, which stems from its low amplitude.

    Following Huber et al., we may also fix ω=ω0 and plot the linear response as a function of F. The response turns out to be single-valued over a large range of driving strengths. Using a log scale for the x-axis:

    julia
    fixed = (α => 1., ω0 => 1.0, γ => 1e-2, ω => 1)   # fixed parameters
+swept = F => 10 .^ range(-6, -1, 200)           # range of parameter values
+result = get_steady_states(harmonic_eq, swept, fixed)
+
+plot(
+  plot(result, "sqrt(u1^2 + v1^2)", xscale=:log),
+  plot_linear_response(result, x, branch=1, Ω_range=range(0.9,1.1,300), logscale=true, xscale=:log),
+  size=(600, 250), margin=3mm
+)

    We see that for low F, quasi-linear behaviour with a single Lorentzian response occurs, while for larger F, two peaks form in the noise response. The two peaks are strongly unequal in magnitude, which is an example of internal squeezing (See supplemental material of Huber et al.).

    + + + + \ No newline at end of file diff --git a/v0.10.7/tutorials/steady_states.html b/v0.10.7/tutorials/steady_states.html new file mode 100644 index 00000000..d129411f --- /dev/null +++ b/v0.10.7/tutorials/steady_states.html @@ -0,0 +1,77 @@ + + + + + + Finding the staedy states of a Duffing oscillator | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

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    Finding the staedy states of a Duffing oscillator ​

    Here we show the workflow of HarmonicBalance.jl on a simple example - the driven Duffing oscillator. The equation of motion for the displacement x(t) reads

    x¨(t)+γx˙(t)+ω02x(t)⏟damped harmonic oscillator+αx(t)3⏟Duffing coefficient=Fcos⁡(ωt)⏟periodic drive

    In general, there is no analytical solution to the differential equation. Fortunately, some harmonics are more important than others. By truncating the infinite-dimensional Fourier space to a set of judiciously chosen harmonics, we may obtain a soluble system. For the Duffing resonator, we can well try to only consider the drive frequency ω. To implement this, we use the harmonic ansatz

    x(t)=Ucos⁡(ωt)+Vsin⁡(ωt),

    which constraints the spectrum of x(t) to a single harmonic. Fixing the quadratures U and V to be constant then reduces the differential equation to two coupled cubic polynomial equations (for more details on this step, see the appendices in the white paper). Finding the roots of coupled polynomials is in general very hard. We here apply the method of homotopy continuation, as implemented in HomotopyContinuation.jl which is guaranteed to find the complete set of roots.

    First we need to declare the symbolic variables (the excellent Symbolics.jl is used here).

    julia
    using HarmonicBalance
+@variables α ω ω0 F γ t x(t) # declare constant variables and a function x(t)

    Next, we have to input the equations of motion. This will be stored as a DifferentialEquation. The input needs to specify that only x is a mathematical variable, the other symbols are parameters:

    julia
    diff_eq = DifferentialEquation(d(x,t,2) + ω0^2*x + α*x^3 + γ*d(x,t) ~ F*cos(ω*t), x)
    System of 1 differential equations
+Variables:       x(t)
+Harmonic ansatz: x(t) => ;   
+
+Differential(t)(Differential(t)(x(t))) + Differential(t)(x(t))*γ + x(t)*(ω0^2) + (x(t)^3)*α ~ F*cos(t*ω)

    One harmonic ​

    The harmonic ansatz needs to be specified now – we expand x in a single frequency ω.

    julia
    add_harmonic!(diff_eq, x, ω) # specify the ansatz x = u(T) cos(ωt) + v(T) sin(ωt)

    The object diff_eq now contains all the necessary information to convert the differential equation to the algebraic harmonic equations (coupled polynomials in U and V).

    julia
    harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, ω0, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ F
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0

    The variables u1 and v1 were declared automatically to construct the harmonic ansatz. The slow time variable T describes variation of the quadratures on timescales much slower than ω. For a steady state, all derivatives w.r.t T vanish, leaving only algebraic equations to be solved.

    We are ready to start plugging in numbers! Let us find steady states by solving harmonic_eq for numerical parameters. Homotopy continuation is especially suited to solving over a range of parameter values. Here we will solve over a range of driving frequencies ω – these are stored as Pairs{Sym, Vector{Float64}}:

    julia
    varied = ω => range(0.9, 1.2, 100); # range of parameter values
    ω => 0.9:0.0030303030303030303:1.2

    The other parameters we be fixed – these are declared as Pairs{Sym, Float64} pairs:

    julia
    fixed = (α => 1., ω0 => 1.0, F => 0.01, γ => 0.01); # fixed parameters
    (α => 1.0, ω0 => 1.0, F => 0.01, γ => 0.01)

    Now everything is ready to crank the handle. get_steady_states solves our harmonic_eq using the varied and fixed parameters:

    julia
    result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 100 parameter points
+
+Solution branches:   3
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels

    The algorithm has found 3 solution branches in total (out of the hypothetically admissible 32=9). All of these are real – and thefore physically observable – for at least some values of ω. Only 2 branches are stable under infinitesimal perturbations. The "Classes" are boolean labels classifying each solution point, which may be used to select results for plotting.

    We now want to visualize the results. Here we plot the solution amplitude, U2+V2 against the drive frequency ω:

    julia
    plot(result, "sqrt(u1^2 + v1^2)")

    This is the expected response curve for the Duffing equation.

    Using multiple harmonics ​

    In the above section, we truncated the Fourier space to a single harmonic ω – the oscillator was assumed to only oscillate at the drive frequency. However, the Duffing oscillator can exhibit a rich spectrum of harmonics. We can obtain some intuition by treating α perturbatively in the equation of motion, i.e., by solving

    x¨(t)+γx˙(t)+ω02x(t)+ϵαx(t)3=Fcos⁡(ωt)

    for small ϵ. To zeroth order, the response of the system is x0(t)=X0cos⁡(ωt+ϕ0). Expanding x(t)=x0(t)+ϵx1(t), we find that the perturbation x1(t) satisfies to first order

    x¨1(t)+γx˙1(t)[ω02+3αX024]x1(t)=−αX034cos⁡(3ωt+3ϕ0),

    which gives a response of the form x1(t)=X1cos⁡(3ωt+ϕ1). Clearly, the oscillator now responds not only at frequency ω, but also at 3ω! This effect is known as high harmonic generation or more generally frequency conversion. By continuing the procedure to higher orders, we eventually obtain an infinity of harmonics present in the response. In general, there is no analytical solution to such problems.

    We argued that frequency conversion takes place, to first order from ω to 3ω. We can reflect this process by using a extended harmonic ansatz:

    x(t)=U1cos⁡(ωt)+V1sin⁡(ωt)+U2cos⁡(3ωt)+V2sin⁡(3ωt).

    Note that this is not a perturbative treatment! The harmonics ω and 3ω are on the same footing here. This is implemented as

    julia
    add_harmonic!(diff_eq, x, [ω, 3ω]) # specify the two-harmonics ansatz
+harmonic_eq = get_harmonic_equations(diff_eq)
    A set of 4 harmonic equations
+Variables: u1(T), v1(T), u2(T), v2(T)
+Parameters: ω, ω0, γ, α, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt) + u2(T)*cos(3ωt) + v2(T)*sin(3ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*u2(T)*α + (3//2)*u1(T)*(v2(T)^2)*α + (3//2)*u1(T)*v2(T)*v1(T)*α + (3//4)*u1(T)*(v1(T)^2)*α + (3//2)*u1(T)*(u2(T)^2)*α - (3//4)*(v1(T)^2)*u2(T)*α ~ F
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v2(T)*α + (3//4)*(u1(T)^2)*v1(T)*α - (3//2)*u1(T)*v1(T)*u2(T)*α + (3//2)*(v2(T)^2)*v1(T)*α - (3//4)*v2(T)*(v1(T)^2)*α + (3//4)*(v1(T)^3)*α + (3//2)*v1(T)*(u2(T)^2)*α ~ 0//1
+
+Differential(T)(u2(T))*γ + (6//1)*Differential(T)(v2(T))*ω + (3//1)*v2(T)*γ*ω - (9//1)*u2(T)*(ω^2) + u2(T)*(ω0^2) + (1//4)*(u1(T)^3)*α + (3//2)*(u1(T)^2)*u2(T)*α - (3//4)*u1(T)*(v1(T)^2)*α + (3//4)*(v2(T)^2)*u2(T)*α + (3//2)*(v1(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0//1
+
+-(6//1)*Differential(T)(u2(T))*ω + Differential(T)(v2(T))*γ - (9//1)*v2(T)*(ω^2) + v2(T)*(ω0^2) - (3//1)*u2(T)*γ*ω + (3//2)*(u1(T)^2)*v2(T)*α + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v2(T)^3)*α + (3//2)*v2(T)*(v1(T)^2)*α + (3//4)*v2(T)*(u2(T)^2)*α - (1//4)*(v1(T)^3)*α ~ 0//1

    The variables u1, v1 now encode ω and u2, v2 encode 3ω. We see this system is much harder to solve as we now have 4 harmonic variables, resulting in 4 coupled cubic equations. A maximum of 34=81 solutions may appear!

    julia
    result = get_steady_states(harmonic_eq, varied, fixed)
+plot(result, "sqrt(u1^2 + v1^2)")

    For the above parameters (where a perturbative treatment would have been reasonable), the principal response at ω looks rather similar, with a much smaller upconverted component appearing at 3ω:

    julia
    p1=plot(result, "sqrt(u1^2 + v1^2)", legend=false)
+p2=plot(result, "sqrt(u2^2 + v2^2)")
+plot(p1, p2)

    The non-perturbative nature of the ansatz allows us to capture some behaviour which is not a mere extension of the usual single-harmonic Duffing response. Suppose we drive a strongly nonlinear resonator at frequency ω≅ω0/3. Such a drive is far out of resonance, however, the upconverted harmonic 3ω=ω0 is not and may play an important role! Let us try this out:

    julia
    fixed = (α => 10., ω0 => 3, F => 5, γ=>0.01)   # fixed parameters
+varied = ω => range(0.9, 1.4, 100)           # range of parameter values
+result = get_steady_states(harmonic_eq, varied, fixed)
    A steady state result for 100 parameter points
+
+Solution branches:   9
+   of which real:    3
+   of which stable:  2
+
+Classes: stable, physical, Hopf, binary_labels

    Although 9 branches were found in total, only 3 remain physical (real-valued). Let us visualise the amplitudes corresponding to the two harmonics, U12+V12 and U22+V22 :

    julia
    p1 = plot(result, "sqrt(u1^2 + v1^2)", legend=false)
+p2 = plot(result, "sqrt(u2^2 + v2^2)")
+plot(p1, p2)

    The contributions of ω and 3ω are now comparable and the system shows some fairly complex behaviour! This demonstrates how an exact solution within an extended Fourier subspace goes beyond a perturbative treatment.

    + + + + \ No newline at end of file diff --git a/v0.10.7/tutorials/time_dependent.html b/v0.10.7/tutorials/time_dependent.html new file mode 100644 index 00000000..57e6a7ed --- /dev/null +++ b/v0.10.7/tutorials/time_dependent.html @@ -0,0 +1,61 @@ + + + + + + Time-dependent simulations | HarmonicBalance.jl + + + + + + + + + + + + + + + +
    HarmonicBalance.jl

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    Time-dependent simulations ​

    Most of HarmonicBalance.jl is focused on finding and analysing the steady states. Such states contain no information about transient behaviour, which is crucial to answer the following.

    • Given an initial condition, which steady state does the system evolve into?

    • How does the system behave if its parameters are varied in time?

    It is straightforward to evolve the full equation of motion using an ODE solver. However, tracking oscillatory behaviour is computationally expensive.

    In the background, we showed that nonlinear driven systems may be reduced to harmonic equations

    du(T)dT=F¯(u),

    As long as the chosen harmonics constituting u(T) capture the system's behaviour, we may numerically evolve the new effective differential equations instead of the full problem. Since the components of u(T) only vary very slowly (and are constant in a steady state), this is usually vastly more efficient than evolving the full problem.

    Here we primarily demonstrate on the parametrically driven oscillator.

    We start by defining our system.

    julia
    using HarmonicBalance
+@variables ω0 γ λ F θ η α ω t x(t)
+
+eq =  d(d(x,t),t) + γ*d(x,t) + ω0^2*(1 - λ*cos(2*ω*t))*x + α*x^3 + η*d(x,t)*x^2 ~ F*cos(ω*t + θ)
+
+diff_eq = DifferentialEquation(eq, x)
+add_harmonic!(diff_eq, x, ω); # single-frequency ansatz
+
+harmonic_eq = get_harmonic_equations(diff_eq);
    A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, α, γ, λ, ω0, η, θ, F
+
+Harmonic ansatz: 
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*Differential(T)(u1(T))*η + (1//2)*u1(T)*Differential(T)(v1(T))*v1(T)*η + (3//4)*u1(T)*(v1(T)^2)*α - (1//2)*u1(T)*λ*(ω0^2) + (1//4)*(v1(T)^2)*Differential(T)(u1(T))*η + (1//4)*(u1(T)^2)*v1(T)*η*ω + (1//4)*(v1(T)^3)*η*ω ~ F*cos(θ)
+
+Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (1//4)*(u1(T)^2)*Differential(T)(v1(T))*η + (3//4)*(u1(T)^2)*v1(T)*α + (1//2)*u1(T)*v1(T)*Differential(T)(u1(T))*η + (3//4)*Differential(T)(v1(T))*(v1(T)^2)*η + (3//4)*(v1(T)^3)*α + (1//2)*v1(T)*λ*(ω0^2) - (1//4)*(u1(T)^3)*η*ω - (1//4)*u1(T)*(v1(T)^2)*η*ω ~ -F*sin(θ)

    The object harmonic_eq encodes the new effective differential equations.

    We now wish to parse this input into OrdinaryDiffEq.jl and use its powerful ODE solvers. The desired object here is OrdinaryDiffEq.ODEProblem, which is then fed into OrdinaryDiffEq.solve.

    Evolving from an initial condition ​

    Given u(T0), what is u(T) at future times?

    For constant parameters, a HarmonicEquation object can be fed into the constructor of ODEProblem. The syntax is similar to DifferentialEquations.jl :

    julia
    using OrdinaryDiffEq
+x0 = [0.; 0.] # initial condition
+fixed = (ω0 => 1.0, γ => 1e-2, λ => 5e-2, F => 1e-3,  α => 1.0, η => 0.3, θ => 0, ω => 1.0) # parameter values
+
+ode_problem = ODEProblem(harmonic_eq, fixed, x0 = x0, timespan = (0,1000))
    ODEProblem with uType Vector{Float64} and tType Int64. In-place: true
+timespan: (0, 1000)
+u0: 2-element Vector{Float64}:
+ 0.0
+ 0.0

    OrdinaryDiffEq.jl takes it from here - we only need to use solve.

    julia
    time_evo = solve(ode_problem, saveat=1.0);
+plot(time_evo, ["u1", "v1"], harmonic_eq)

    Running the above code with x0 = [0.2, 0.2] gives the plots

    julia
    x0 = [0.2; 0.2] # initial condition
+ode_problem = remake(ode_problem, u0 = x0)
+time_evo = solve(ode_problem, saveat=1.0);
+plot(time_evo, ["u1", "v1"], harmonic_eq)

    Let us compare this to the steady state diagram.

    julia
    varied = ω => range(0.9, 1.1, 100)
+result = get_steady_states(harmonic_eq, varied, fixed)
+plot(result, "sqrt(u1^2 + v1^2)")

    Clearly when evolving from x0 = [0.,0.], the system ends up in the low-amplitude branch 2. With x0 = [0.2, 0.2], the system ends up in branch 3.

    Adiabatic parameter sweeps ​

    Experimentally, the primary means of exploring the steady state landscape is an adiabatic sweep one or more of the system parameters. This takes the system along a solution branch. If this branch disappears or becomes unstable, a jump occurs.

    The object ParameterSweep specifies a sweep, which is then used as an optional sweep keyword in the ODEProblem constructor.

    julia
    sweep = ParameterSweep(ω => (0.9,1.1), (0, 2e4))
    ParameterSweep(Dict{Num, Function}(ω => TimeEvolution.var"#f#1"{Tuple{Float64, Float64}, Float64, Int64}((0.9, 1.1), 20000.0, 0)))

    The sweep linearly interpolates between ω=0.9 at time 0 and ω=1.1 at time 2e4. For earlier/later times, ω is constant.

    Let us now define a new ODEProblem which incorporates sweep and again use solve:

    julia
    ode_problem = ODEProblem(harmonic_eq, fixed, sweep=sweep, x0=[0.1;0.0], timespan=(0, 2e4))
+time_evo = solve(ode_problem, saveat=100)
+plot(time_evo, "sqrt(u1^2 + v1^2)", harmonic_eq)

    We see the system first evolves from the initial condition towards the low-amplitude steady state. The amplitude increases as the sweep proceeds, with a jump occurring around ω=1.08 (i.e., time 18000).

    + + + + \ No newline at end of file diff --git a/versions.js b/versions.js index 73d7a1c6..dc9dde3d 100644 --- a/versions.js +++ b/versions.js @@ -9,5 +9,5 @@ var DOC_VERSIONS = [ "v0.4", "dev", ]; -var DOCUMENTER_NEWEST = "v0.10.6"; +var DOCUMENTER_NEWEST = "v0.10.7"; var DOCUMENTER_STABLE = "stable";
    HarmonicBalance.jl

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    Krylov-Bogoliubov Averaging Method ​

    The Krylov-Bogoliubov averaging method is an alternative high-frequency expansion technique used to analyze dynamical systems. Unlike the Harmonic Balance method, which is detailed in the background section, the Krylov-Bogoliubov method excels in computing higher orders in 1/ω, enabling the capture of faster dynamics within a system.

    Purpose and Advantages ​

    The primary advantage of the Krylov-Bogoliubov method lies in its ability to delve deeper into high-frequency components, allowing a more comprehensive understanding of fast dynamical behaviors. By leveraging this technique, one can obtain higher-order approximations that shed light on intricate system dynamics.

    However, it's essential to note a limitation: this method cannot handle multiple harmonics within a single variable, unlike some other high-frequency expansion methods.

    Usage ​

    To compute the Krylov-Bogoliubov averaging method within your system, utilize the function get_krylov_equations. This function is designed specifically to implement the methodology and derive the equations necessary to analyze the system dynamics using this technique.

    Function Reference ​

    # HarmonicBalance.KrylovBogoliubov.get_krylov_equations — Function.
    julia
    get_krylov_equations(
+    diff_eom::DifferentialEquation;
+    order,
+    fast_time,
+    slow_time
+)

    Apply the Krylov-Bogoliubov averaging method to a specific order to obtain a set of ODEs (the slow-flow equations) governing the harmonics of diff_eom.

    The harmonics evolve in slow_time, the oscillating terms themselves in fast_time. If no input is used, a variable T is defined for slow_time and fast_time is taken as the independent variable of diff_eom.

    Krylov-Bogoliubov averaging method can be applied up to order = 2.

    Example

    julia
    julia> @variables t, x(t), ω0, ω, F;
+
+# enter the simple harmonic oscillator
+julia> diff_eom = DifferentialEquation( d(x,t,2) + ω0^2 * x ~ F *cos(ω*t), x);
+
+# expand x in the harmonic ω
+julia> add_harmonic!(diff_eom, x, ω);
+
+# get equations for the harmonics evolving in the slow time T to first order
+julia> harmonic_eom = get_krylov_equations(diff_eom, order = 1)
+
+A set of 2 harmonic equations
+Variables: u1(T), v1(T)
+Parameters: ω, F, ω0
+
+Harmonic ansatz:
+xˍt(t) =
+x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)
+
+Harmonic equations:
+
+((1//2)*(ω^2)*v1(T) - (1//2)*(ω0^2)*v1(T)) / ω ~ Differential(T)(u1(T))
+
+((1//2)*(ω0^2)*u1(T) - (1//2)*F - (1//2)*(ω^2)*u1(T)) / ω ~ Differential(T)(v1(T))

    source


    For further information and a detailed understanding of this method, refer to Krylov-Bogoliubov averaging method on Wikipedia.

    Purpose and Advantages ​

    • Citation ​

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