Fix formatting in limit_cycles.md and methods.jl documentation

docs/src/tutorials/limit_cycles.md

@@ -62,8 +62,8 @@ Here we reconstruct the results of [Zambon et al., Phys Rev. A 102, 023526 (2020
 ```@example lc
 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)]
+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])
 ```
 
@@ -82,8 +82,7 @@ fixed = (
     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)
+    η => -0.08     # pumping leaking to site 2  (F2 = ηF1)
 )
 varied = F0 => range(0.002, 0.03, 50)
 

src/methods.jl

@@ -9,7 +9,7 @@ abstract type HarmonicBalanceMethod end
     TotalDegree
 
 The Total Degree homotopy method. Performs a homotopy ``H(x, t) = γ t G(x) + (1-t) F(x)``
-from the trivial polynomial system ``xᵢ^dᵢ +aᵢ`` with the maximal degree ``dᵢ`` determined
+from the trivial polynomial system ``xᵢ^{dᵢ} +aᵢ`` with the maximal degree ``dᵢ`` determined
 by the [Bezout bound](https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem). The method
 guarantees to find all solutions, however, it comes with a high computational cost. See
 [HomotopyContinuation.jl](https://www.juliahomotopycontinuation.org/guides/totaldegree/)