build based on a9607d4

dev/.documenter-siteinfo.json

@@ -1 +1 @@
-{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-06-10T09:35:34","documenter_version":"1.4.1"}}
+{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-06-10T09:38:39","documenter_version":"1.4.1"}}

dev/background/harmonic_balance/index.html

@@ -37,4 +37,4 @@
 	\\
 	\frac{dv_2}{dT} &= \frac{1}{6 \omega_d} \left[ \left(9\omega_d^2 - \omega_0^2\right) {u_2} - \frac{\alpha}{4} \left( u_1^3 + 3 u_2^3 + 6 u_1^2 u_2 - 3 v_1^2 u_1 + 3 v_2^2 u_2 + 6 v_1^2 u_2\right) \right] \:.
 	\end{split}
-	\end{align}\]</p><p>In contrast to the single-frequency ansatz [Eqs. \eqref{eq:ansatz1}],  we now have 4 equations of order 3, allowing up to <span>$3^4=81$</span> 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 <span>$3 \omega_d$</span> component is significant. Such solutions appear, e.g., for <span>$\omega_d \approx \omega_0 / 3$</span> where the generated <span>$3 \omega_d$</span> harmonic is close to the natural resonant frequency. See the <a href="../../examples/simple_Duffing/#Duffing">examples</a> for numerical results.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="../stability_response/">Stability and linear response »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Monday 10 June 2024 09:35">Monday 10 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+	\end{align}\]</p><p>In contrast to the single-frequency ansatz [Eqs. \eqref{eq:ansatz1}],  we now have 4 equations of order 3, allowing up to <span>$3^4=81$</span> 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 <span>$3 \omega_d$</span> component is significant. Such solutions appear, e.g., for <span>$\omega_d \approx \omega_0 / 3$</span> where the generated <span>$3 \omega_d$</span> harmonic is close to the natural resonant frequency. See the <a href="../../examples/simple_Duffing/#Duffing">examples</a> for numerical results.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="../stability_response/">Stability and linear response »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Monday 10 June 2024 09:38">Monday 10 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/background/stability_response/index.html

@@ -26,4 +26,4 @@
 \end{multline}\]</p><p>Keeping in mind that <span>$L(x)_{x_0, \gamma} = L(x + \Delta)_{x_0 + \Delta, \gamma}$</span> and the normalization <span>$\delta \hat{u}_{i,j}^2 + \delta \hat{v}_{i,j}^2 = 1$</span>, we can rewrite this as</p><p class="math-container">\[\begin{equation}
 |\chi [\delta x _i](\tilde{\omega})|^2 = \sum_{j=1}^{M_i} \left( 1 + \alpha_{i,j} \right) L(\tilde{\omega})_{\omega_{i,j} - \text{Im}[\lambda], \text{Re}[\lambda]}
 + \left( 1 - \alpha_{i,j} \right) L(\tilde{\omega})_{\omega_{i,j} + \text{Im}[\lambda], \text{Re}[\lambda]}
-\end{equation}\]</p><p>where </p><p class="math-container">\[\alpha_{i,j} = 2\left( \text{Im}[\delta \hat{u}_{i,j}] \text{Re}[\delta \hat{v}_{i,j}] - \text{Re}[\delta \hat{u}_{i,j}] \text{Im}[\delta \hat{v}_{i,j}] \right)\]</p><p>The above solution applies to every eigenvalue <span>$\lambda$</span> of the Jacobian. It is now clear that the linear response function <span>$\chi [\delta x _i](\tilde{\omega})$</span> contains for each eigenvalue <span>$\lambda_r$</span> and harmonic <span>$\omega_{i,j}$</span> : </p><ul><li>A Lorentzian centered at <span>$\omega_{i,j}-\text{Im}[\lambda_r]$</span> with amplitude <span>$1 + \alpha_{i,j}^{(r)}$</span> </li><li>A Lorentzian centered at <span>$\omega_{i,j}+\text{Im}[\lambda_r]$</span> with amplitude <span>$1 - \alpha_{i,j}^{(r)}$</span> </li></ul><p><em>Sidenote:</em> As <span>$J$</span> a real matrix, there is an eigenvalue <span>$\lambda_r^*$</span> for each <span>$\lambda_r$</span>. The maximum number of peaks in the linear response is thus equal to the dimensionality of <span>$\mathbf{u}(T)$</span>.</p><p>The linear response of the system in the state <span>$\mathbf{u}_0$</span> is thus fully specified by the complex eigenvalues and eigenvectors of <span>$J(\mathbf{u}_0)$</span>. In HarmonicBalance.jl, the module <a href="../../manual/linear_response/#linresp_man">LinearResponse</a> creates a set of plottable <a href="../../manual/linear_response/#HarmonicBalance.LinearResponse.Lorentzian"><code>Lorentzian</code></a> objects to represent this.</p><p><a href="../../examples/linear_response/#linresp_ex">Check out this example</a> of the linear response module of HarmonicBalance.jl</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../harmonic_balance/">« The method of harmonic balance</a><a class="docs-footer-nextpage" href="../limit_cycles/">Limit cycles »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Monday 10 June 2024 09:35">Monday 10 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{equation}\]</p><p>where </p><p class="math-container">\[\alpha_{i,j} = 2\left( \text{Im}[\delta \hat{u}_{i,j}] \text{Re}[\delta \hat{v}_{i,j}] - \text{Re}[\delta \hat{u}_{i,j}] \text{Im}[\delta \hat{v}_{i,j}] \right)\]</p><p>The above solution applies to every eigenvalue <span>$\lambda$</span> of the Jacobian. It is now clear that the linear response function <span>$\chi [\delta x _i](\tilde{\omega})$</span> contains for each eigenvalue <span>$\lambda_r$</span> and harmonic <span>$\omega_{i,j}$</span> : </p><ul><li>A Lorentzian centered at <span>$\omega_{i,j}-\text{Im}[\lambda_r]$</span> with amplitude <span>$1 + \alpha_{i,j}^{(r)}$</span> </li><li>A Lorentzian centered at <span>$\omega_{i,j}+\text{Im}[\lambda_r]$</span> with amplitude <span>$1 - \alpha_{i,j}^{(r)}$</span> </li></ul><p><em>Sidenote:</em> As <span>$J$</span> a real matrix, there is an eigenvalue <span>$\lambda_r^*$</span> for each <span>$\lambda_r$</span>. The maximum number of peaks in the linear response is thus equal to the dimensionality of <span>$\mathbf{u}(T)$</span>.</p><p>The linear response of the system in the state <span>$\mathbf{u}_0$</span> is thus fully specified by the complex eigenvalues and eigenvectors of <span>$J(\mathbf{u}_0)$</span>. In HarmonicBalance.jl, the module <a href="../../manual/linear_response/#linresp_man">LinearResponse</a> creates a set of plottable <a href="../../manual/linear_response/#HarmonicBalance.LinearResponse.Lorentzian"><code>Lorentzian</code></a> objects to represent this.</p><p><a href="../../examples/linear_response/#linresp_ex">Check out this example</a> of the linear response module of HarmonicBalance.jl</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../harmonic_balance/">« The method of harmonic balance</a><a class="docs-footer-nextpage" href="../limit_cycles/">Limit cycles »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Monday 10 June 2024 09:38">Monday 10 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/examples/limit_cycles/index.html

@@ -8,4 +8,4 @@
 
 Solution branches:   100
    of which real:    4
-   of which stable:  4 </code></pre><p>The results show a fourfold <a href="../../background/limit_cycles/#degeneracies">degeneracy of solutions</a>. The automatically created solution class <code>unique_cycle</code> filters the degeneracy out,</p><pre><code class="language-julia hljs">plot(result, ω_lc)</code></pre><p><img src="../../assets/limit_cycles/vdp_degenerate.png" alt="fig1"/></p><pre><code class="language-julia hljs">plot(result, ω_lc, class=&quot;unique_cycle&quot;)</code></pre><p><img src="../../assets/limit_cycles/vdp_nondegenerate.png" alt="fig2"/></p><h2 id="Driven-system-coupled-Duffings"><a class="docs-heading-anchor" href="#Driven-system-coupled-Duffings">Driven system - coupled Duffings</a><a id="Driven-system-coupled-Duffings-1"></a><a class="docs-heading-anchor-permalink" href="#Driven-system-coupled-Duffings" title="Permalink"></a></h2><p>Under construction, see Chapter 6.2.2 of <a href="https://www.research-collection.ethz.ch/handle/20.500.11850/589190">Jan&#39;s thesis</a></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../parametron/">« Parametrically driven Duffing resonator: 1D and 2D plots</a><a class="docs-footer-nextpage" href="../../manual/entering_eom/">Entering equations of motion »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Monday 10 June 2024 09:35">Monday 10 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+   of which stable:  4 </code></pre><p>The results show a fourfold <a href="../../background/limit_cycles/#degeneracies">degeneracy of solutions</a>. The automatically created solution class <code>unique_cycle</code> filters the degeneracy out,</p><pre><code class="language-julia hljs">plot(result, ω_lc)</code></pre><p><img src="../../assets/limit_cycles/vdp_degenerate.png" alt="fig1"/></p><pre><code class="language-julia hljs">plot(result, ω_lc, class=&quot;unique_cycle&quot;)</code></pre><p><img src="../../assets/limit_cycles/vdp_nondegenerate.png" alt="fig2"/></p><h2 id="Driven-system-coupled-Duffings"><a class="docs-heading-anchor" href="#Driven-system-coupled-Duffings">Driven system - coupled Duffings</a><a id="Driven-system-coupled-Duffings-1"></a><a class="docs-heading-anchor-permalink" href="#Driven-system-coupled-Duffings" title="Permalink"></a></h2><p>Under construction, see Chapter 6.2.2 of <a href="https://www.research-collection.ethz.ch/handle/20.500.11850/589190">Jan&#39;s thesis</a></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../parametron/">« Parametrically driven Duffing resonator: 1D and 2D plots</a><a class="docs-footer-nextpage" href="../../manual/entering_eom/">Entering equations of motion »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Monday 10 June 2024 09:38">Monday 10 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

dev/examples/linear_response/index.html

@@ -29,4 +29,4 @@
 plot(result, "sqrt(u1^2 + v1^2)", xscale=:log)
 
 plot_linear_response(result, x, branch=1, 
-    Ω_range=LinRange(0.9,1.1,300), order=1, logscale=true, xscale=:log)</code></pre><img style="display: block; margin: 0 auto;" src="../../assets/linear_response/nonlin_F_noise.png" alignment="left" \><p>We see that for low <span>$F$</span>, quasi-linear behaviour with a single Lorentzian response occurs, while for larger <span>$F$</span>, two peaks form in the noise response. The two peaks are strongly unequal in magnitude, which is an example of internal squeezing.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../simple_Duffing/">« Introduction: the Duffing oscillator</a><a class="docs-footer-nextpage" href="../time_dependent/">Introduction: time-dependent simulations »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Monday 10 June 2024 09:35">Monday 10 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+    Ω_range=LinRange(0.9,1.1,300), order=1, logscale=true, xscale=:log)</code></pre><img style="display: block; margin: 0 auto;" src="../../assets/linear_response/nonlin_F_noise.png" alignment="left" \><p>We see that for low <span>$F$</span>, quasi-linear behaviour with a single Lorentzian response occurs, while for larger <span>$F$</span>, two peaks form in the noise response. The two peaks are strongly unequal in magnitude, which is an example of internal squeezing.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../simple_Duffing/">« Introduction: the Duffing oscillator</a><a class="docs-footer-nextpage" href="../time_dependent/">Introduction: time-dependent simulations »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Monday 10 June 2024 09:38">Monday 10 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>