This is still a bit mysterious to me… I am not sure I see the point.
You dont need to know the period for the conventional methods to work, actually the period is an unknown. It yields a BVP problem with a phase condition. The combination of Newton and continuation is a “basic” component of numerical bifurcation (see periodic orbits) and is implemented in BifurcationKit.jl
Plus, in orthogonal collocation, you can use mesh adaptation. Close to homoclinic orbits, the Fourier expansion will not yield good results.
Perhaps I should restate my original need. I would like to know whether the ODE ends up giving an oscillatory equilibrium. The implementation callback = TerminateSteadyState() helps me by telling whether equilibrium has been reached. If eq has fewer cycles than soln, then an equilibrium had been reached and the maximum time point of eq tells me when. However, this implementation works for equilibria that do not change, like:
Can something similar be done with oscillatory equilibria?
Thanks
The oscillations appear at kappa = 2.004616330703016e7 (Hopf bifurcation) with state = [1.2295081967213114e6, 389184.48612039164, 9.496101461337554e7, 206056.3353121641, 59463.21329809305, 2.308619658134151e8].
julia> hopf
SuperCritical - Hopf bifurcation point at κ ≈ 2.004616330703016e7.
Frequency ω ≈ 0.06391634078691726
Period of the periodic orbit ≈ 98.30327002176668
Normal form z⋅(iω + a⋅δp + b⋅|z|²):
(a = 5.967587852411348e-11 + 1.655914621958718e-11im, b = -3.504679044424272e-19 + 2.9623334806795282e-18im)
Sorry for the late reply. May I ask you how did you implement hopf? Is there a gentle introduction to bifurcation apart from the link you gave? Thank you
For the code, I used BifurcationKit.jl. I will try to post the code when I find some time
For books, maybe
Hirsch, Morris W., Stephen Smale, Robert L. Devaney, and Morris W. Hirsch. Differential Equations, Dynamical Systems, and an Introduction to Chaos. 2nd ed. Pure and Applied Mathematics; a Series of Monographs and Textbooks, v. 60. San Diego, CA: Academic Press, 2004.
Kuznetsov, Yuri A. Elements of Applied Bifurcation Theory, Second Edition, n.d.
