Branch switching

The precise definition of the methods are given in Branch switching (branch point) and Branch switching (Hopf point).

• Branch switching
• • Branch switching from simple branch point to equilibria
  • Branch switching from non simple branch point to equilibria
  • Branch switching from Hopf point to periodic orbits
  • From Hopf-Hopf (HH) point to Fold / Hopf curve

Branch switching from simple branch point to equilibria

You can perform automatic branch switching by calling continuation with the following options:

continuation(br::ContResult, ind_bif::Int, optionsCont::ContinuationPar; kwargs...)

where br is a branch computed after a call to continuation with detection of bifurcation points enabled. This call computes the branch bifurcating from the ind_bifth bifurcation point in br. An example of use is provided in 2d generalized Bratu–Gelfand problem.

See Branch switching (branch point) precise method definition

Branch switching from non simple branch point to equilibria

We provide an automatic branch switching method in this case. The method is to first compute the reduced equation (see Non-simple branch point) and use it to compute the nearby solutions. These solutions are seeded as initial guess for continuation. Hence, you can perform automatic branch switching by calling continuation with the following options:

continuation(br::ContResult, ind_bif::Int, optionsCont::ContinuationPar; kwargs...)

An example of use is provided in 2d generalized Bratu–Gelfand problem.

See Branch switching (branch point) for the precise method definition

Branch switching from Hopf point to periodic orbits

In order to compute the bifurcated branch of periodic solutions at a Hopf bifurcation point, you need to choose a method to compute periodic orbits among:

• Periodic orbits based on orthogonal collocation

Once you have decided which method to use, you use the following call:

continuation(br::ContResult, ind_HOPF::Int, _contParams::ContinuationPar,	prob::AbstractPeriodicOrbitProblem ;	δp = nothing, ampfactor = 1, kwargs...)

We refer to continuation for more information about the arguments. Here, we just say a few words about how we can specify prob::AbstractPeriodicOrbitProblem.

• For Periodic orbits based on orthogonal collocation, you can pass PeriodicOrbitOCollProblem(M, m) where M is the number of times slices in the periodic orbit and m is the degree of the collocation polynomials.

From Hopf-Hopf (HH) point to Fold / Hopf curve

We provide an automatic branch switching method in this case (see for example Neuron model). Hence, you can perform automatic branch switching by calling continuation with the following options:

continuation(br::ContResult, ind_HH::Int,	options_cont::ContinuationPar = br.contparams;	detailed = Val(false), # MUST BE PASSED	alg = getalg(br),	δp = nothing, ampfactor::Real = 1,	nev = options_cont.nev,	detect_codim2_bifurcation::Int = 0,	start_with_eigen = false,	scaleζ = norm,	kwargs...)

where ind_HH is the index of the HH point in br. Note that the HH has been detected during Hopf continuation. Calling the above method thus switches from Hopf continuation to another Hopf branch automatically with the same parameter axis.

Check the docs of Fold / Hopf Continuation and particularly Setting the jacobian for improving the speed of computation for large scale systems.