Autonomous electronic circuit (aBS from BT)

• Autonomous electronic circuit (aBS from BT)
• • Codimension 2 bifurcations
  • Branch of homoclinic orbits with Orthogonal Collocation
  • Branch of homoclinic orbits with Multiple Shooting
  • References

The following model is taken from ^[Freire]:

\[\left\{\begin{aligned} & r \dot{x}=-\nu x+\beta(y-x)-A_3 x^3+B_3(y-x)^3+\epsilon \\ & \dot{y}=-\beta(y-x)-z-B_3(y-x)^3 \\ & \dot{z}=y \end{aligned}\right.\]

This is an example of computing the curve of homoclinics from a Bogdanov-Takens bifurcation point. It is easy to encode the ODE as follows

using Revise, Plots
using Parameters, Setfield, LinearAlgebra, Test, ForwardDiff
using BifurcationKit, Test
using HclinicBifurcationKit
const BK = BifurcationKit

recordFromSolution(x, p) = (x = x[1], y = x[2])

function freire!(dz, u, p, t)
	@unpack ν, β, A₃, B₃, r, ϵ = p
	x, y, z = u
	dz[1] = (-ν*x + β*(y-x) - A₃*x^3 + B₃*(y-x)^3 + ϵ)/r
	dz[2] =	-β*(y-x) - z - B₃*(y-x)^3
	dz[3] = y
	dz
end

freire(z, p) = freire!(similar(z), z, p, 0)
par_freire = (ν = -0.75, β = -0.1, A₃ = 0.328578, B₃ = 0.933578, r = 0.6, ϵ = 0.01)
z0 = [0.7,0.3,0.1]
z0 = zeros(3)
prob = BK.BifurcationProblem(freire, z0, par_freire, (@lens _.β); record_from_solution = recordFromSolution)

We first compute the branch of equilibria

opts_br = ContinuationPar(p_min = -1.4, p_max = 2.8, ds = 0.001, dsmax = 0.05, n_inversion = 6, detect_bifurcation = 3, max_bisection_steps = 25, nev = 3)

br = continuation(prob, PALC(tangent = Bordered()), opts_br, normC = norminf)

scene = plot(br, plotfold=true)

With detailed information:

br

 ┌─ Curve type: EquilibriumCont
 ├─ Number of points: 55
 ├─ Type of vectors: Vector{Float64}
 ├─ Parameter β starts at -0.1, ends at -1.4
 ├─ Algo: PALC
 └─ Special points:

If `br` is the name of the branch,
ind_ev = index of the bifurcating eigenvalue e.g. `br.eig[idx].eigenvals[ind_ev]`

- #  1,     hopf at β ≈ -0.00049754 ∈ (-0.00049760, -0.00049754), |δp|=6e-08, [converged], δ = (-2, -2), step =  10, eigenelements in eig[ 11], ind_ev =   3
- #  2,       bp at β ≈ +0.65519875 ∈ (+0.65519875, +0.65520059), |δp|=2e-06, [converged], δ = (-1,  0), step =  21, eigenelements in eig[ 22], ind_ev =   1
- #  3,     hopf at β ≈ +0.64972077 ∈ (+0.64972077, +0.64973229), |δp|=1e-05, [converged], δ = ( 2,  2), step =  22, eigenelements in eig[ 23], ind_ev =   2
- #  4,     hopf at β ≈ -0.04305104 ∈ (-0.04305104, -0.04304897), |δp|=2e-06, [converged], δ = (-2, -2), step =  34, eigenelements in eig[ 35], ind_ev =   2
- #  5, endpoint at β ≈ -1.40000000,                                                                      step =  54

Codimension 2 bifurcations

sn_br = continuation(br, 2, (@lens _.ν), ContinuationPar(opts_br, detect_bifurcation = 1, save_sol_every_step = 1, dsmax = 0.01, max_steps = 80) ;
	alg = PALC(),
	detect_codim2_bifurcation = 2,
	start_with_eigen = true,
	update_minaug_every_step = 1,
	bothside = true,
	)

hopf_br = continuation(br, 4, (@lens _.ν), ContinuationPar(opts_br, detect_bifurcation = 1, save_sol_every_step = 1, max_steps = 140, dsmax = 0.02, n_inversion = 6),
	detect_codim2_bifurcation = 2,
	start_with_eigen = true,
	update_minaug_every_step = 1,
	bothside = true,
	)

plot(sn_br, hopf_br, ylims = (-0.1, 1.25))

Branch of homoclinic orbits with Orthogonal Collocation

Plotting function:

function plotHom(x,p;k...)
	𝐇𝐨𝐦 = p.prob
	par0 = set(BK.getparams(𝐇𝐨𝐦), BK.getlens(𝐇𝐨𝐦), x.x[end][1])
	par0 = set(par0, p.lens, p.p)
	sol = get_homoclinic_orbit(𝐇𝐨𝐦, x, par0)
	m = (𝐇𝐨𝐦.bvp isa PeriodicOrbitOCollProblem && 𝐇𝐨𝐦.bvp.meshadapt) ? :d : :none
	plot!(sol.t, sol[:,:]',subplot=3, markersize = 1, marker=m)
end

plotHom (generic function with 1 method)

Branching from BT point

# Bogdanov-Takens bifurcation point
btpt = get_normal_form(sn_br, 2; nev = 3, autodiff = false)

# curve of homoclinics
br_hom_c = continuation(
			prob,
			btpt,
			# we use mesh adaptation
			PeriodicOrbitOCollProblem(50, 3; meshadapt = false, K = 200),
			PALC(tangent = Bordered()),
			setproperties(opts_br, max_steps = 30, save_sol_every_step = 1, dsmax = 1e-2, plot_every_step = 1, p_min = -1.01, ds = 0.001, detect_event = 2, detect_bifurcation = 0);
	verbosity = 0, plot = false,
	ϵ0 = 1e-5, amplitude = 2e-3,
	# freeparams = ((@lens _.T), (@lens _.ϵ1),)
	# freeparams = ((@lens _.T), (@lens _.ϵ0)),
	freeparams = ((@lens _.ϵ0), (@lens _.ϵ1)),
	normC = norminf,
	update_every_step = 4,
	)
plot(sn_br, hopf_br, ylims = (0, 1.25), branchlabel = ["SN", "HOPF"])
plot!(br_hom_c, branchlabel = "Hom")

Branch of homoclinic orbits with Multiple Shooting

using DifferentialEquations
probsh = ODEProblem(freire!, copy(z0), (0., 1000.), par_freire; abstol = 1e-12, reltol = 1e-10)

optn_hom = NewtonPar(verbose = true, tol = 1e-10, max_iterations = 7)
optc_hom = ContinuationPar(newton_options = optn_hom, ds = 1e-4, dsmin = 1e-6, dsmax = 1e-3, plot_every_step = 1,max_steps = 10, detect_bifurcation = 0, save_sol_every_step = 1)

br_hom_sh = continuation(
			prob,
			btpt,
			ShootingProblem(12, probsh, Rodas5P(); parallel = true, abstol = 1e-13, reltol = 1e-12),
			PALC(tangent = Bordered()),
			setproperties(optc_hom, max_steps = 200, save_sol_every_step = 1, ds = 1e-3, dsmax = 3e-2, plot_every_step = 50, detect_event = 2, a = 0.9, p_min = -1.01);
	verbosity = 1, plot = true,
	ϵ0 = 1e-6, amplitude = 2e-2,
	update_every_step = 2,
	# freeparams = ((@lens _.T), (@lens _.ϵ1),),
	# freeparams = ((@lens _.T), (@lens _.ϵ0)),
	# freeparams = ((@lens _.ϵ0), (@lens _.ϵ1)),
	freeparams = ((@lens _.T),),
	normC = norminf,
	plot_solution = plotHom,
	maxT = 45.,
	callback_newton = BK.cbMaxNorm(1e1),
	)
title!("")

References

^[Freire] :> Freire, E., A.J. Rodríguez-Luis, E. Gamero, and E. Ponce. “A Case Study for Homoclinic Chaos in an Autonomous Electronic Circuit.” Physica D: Nonlinear Phenomena 62, no. 1–4 (January 1993): 230–53. https://doi.org/10.1016/0167-2789(93)90284-8.