Humphries model (codim 2, periodic orbit)

• Humphries model (codim 2, periodic orbit)
• • Continuation and codim 1 bifurcations
  • Continuation of Hopf point
  • Branch of periodic orbits
  • References

Consider the model ^[Hum] as an example of state-dependent delays

\[x^{\prime}(t)=-\gamma x(t)-\kappa_1 x\left(t-a_1-c x(t)\right)-\kappa_2 x\left(t-a_2-c x(t)\right)\]

Continuation and codim 1 bifurcations

We first instantiate the model

using Revise, DDEBifurcationKit, Plotsusing BifurcationKitconst BK = BifurcationKitfunction humpriesVF(x, xd, p)   (;κ1, κ2, γ, a1, a2, c) = p   [      -γ * x[1] - κ1 * xd.u[1][1] - κ2 * xd.u[2][1]   ]endfunction delaysF(x, par)   [      par.a1 + par.c * x[1],      par.a2 + par.c * x[1],   ]endpars = (κ1=0., κ2=2.3, a1=1.3, a2=6, γ=4.75, c=1.)x0 = zeros(1)prob = SDDDEBifProblem(humpriesVF, delaysF, x0, pars, (@optic _.κ1))

┌─ State-dependent delays Bifurcation Problem with uType Vector{Float64}
├─ Inplace:  false
├─ Dimension: 1
└─ Parameter: κ1

We then compute the branch

optn = NewtonPar(eigsolver = DDE_DefaultEig())opts = ContinuationPar(p_max = 13., p_min = 0., newton_options = optn, ds = -0.01, detect_bifurcation = 3, nev = 3, )br = continuation(prob, PALC(), opts; verbosity = 0, bothside = true)

 ┌─ Curve type: EquilibriumCont
 ├─ Number of points: 100
 ├─ Type of vectors: Vector{Float64}
 ├─ Parameter κ1 starts at 13.0, ends at 0.0
 ├─ Algo: PALC [Secant]
 └─ Special points:

- #  1, endpoint at κ1 ≈ +13.00000000,                                                                     step =   0
- #  2,     hopf at κ1 ≈ +4.28020573 ∈ (+4.27136690, +4.28020573), |δp|=9e-03, [converged], δ = (-2, -2), step =  62
- #  3, endpoint at κ1 ≈ +0.00000000,                                                                     step =  99

and plot it

scene = plot(br)

Continuation of Hopf point

We follow the Hopf points in the parameter plane $(\kappa_1,\kappa_2)$. We tell the solver to consider br.specialpoint[2] and continue it.

brhopf = continuation(br, 2, (@optic _.κ2),         ContinuationPar(br.contparams, detect_bifurcation = 2, dsmax = 0.04, max_steps = 230, p_max = 5., p_min = -1.,ds = -0.02);         detect_codim2_bifurcation = 2,         bothside = true,         start_with_eigen = true)scene = plot(brhopf, vars = (:κ1, :κ2))

Branch of periodic orbits

We compute the branch of periodic orbits from the Hopf bifurcation points using orthogonal collocation. We use a lot of time sections $N_{tst}=200$ to have enough precision to resolve the sophisticated branch of periodic solutions especially near the first Fold point around $\kappa_1\approx 10$.

# continuation parametersopts_po_cont = ContinuationPar(dsmax = 0.1, ds = 0.001, dsmin = 1e-4, p_max = 12., max_steps = 3000, detect_bifurcation = 3, plot_every_step = 20, tol_stability = 4e-3)# arguments for periodic orbitsargs_po = (	plot_solution = (x, p; k...) -> begin		xtt = BK.get_periodic_orbit(p.prob, x, nothing)		plot!(xtt.t, xtt[1,:]; label = "x", k...)		plot!(br; subplot = 1, putspecialptlegend = false)		end,	normC = norminf)probpo = Collocation(30, 5; N = 1, jacobian = BK.DenseAnalytical(), meshadapt = true, K = 400.)br_pocoll = continuation(	br, 2, opts_po_cont,	probpo;	# regular continuation options	alg = PALC(tangent = Bordered()),	args_po...,	δp = 0.05,	callback_newton = BK.cbMaxNorm(10.0),	)scene = plot(br, br_pocoll)

References