Branch switching

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

Branch switching

Summary of branching procedures

We collect in the following table the list of automatic branch switching (aBS) functions. Their detailed explanation follows in this page.

function	ind-th bif. point	Type `T`	description
`continuation(br::ContResult{T}, ind::Int; kw...)`	`:bp`, `:nd`	`EquilibriumCont`	aBS from equilibria to equilibria
`continuation(br::ContResult{T}, ind::Int, lens2::Lens; kw...)`	`:bp`, `:hopf`	`EquilibriumCont`	Fold/Hopf continuation w.r.t. parameters `getlens(br)` and `lens2`
`continuation(br::ContResult{T}, ind::Int; kw...)`	`:bt,:zh,:hh`	`FoldCont,HopfCont`	switch to Fold/Hopf continuation from Hopf/Fold w.r.t. parameters of codim 2 `br`
`continuation(br::ContResult{T}, ind_hopf::Int, ::ContinuationPar, prob::AbstractPeriodicOrbitProblem)`	`:hopf`	`EquilibriumCont`	Branch switching from Hopf point to periodic orbits
`continuation(br::ContResult{T}, ind::Int, kw...)`	`:bp,:pd`	`PeriodicOrbitCont`	Branch switching from Branch / Period-doubling point of periodic orbits to curve of periodic orbits
`continuation(br::ContResult{T}, ind::Int, kw...)`	`:gh,:zh,:hh`	`TwoParamCont`	Branch switching from Bautin / Zero-Hopf/ Hopf-Hopf point to curve of Fold/NS of periodic orbits

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

Simple example

using BifurcationKit, Setfield, Plots

# vector field of transcritical bifurcation
F(x, p) = [x[1] * (p.μ - x[1])]

# parameters of the vector field
par = (μ = -0.2, )

# problem (automatic differentiation)
prob = BifurcationProblem(F, [0.], par, (@lens _.μ); record_from_solution = (x, p) -> x[1])

# compute branch of trivial equilibria (=0) and detect a bifurcation point
opts_br = ContinuationPar(detect_bifurcation = 3)
br = continuation(prob, PALC(), opts_br)

# perform branch switching on one side of the bifurcation point
br1Top = continuation(br, 1, setproperties(opts_br; max_steps = 14) )

# on the other side
br1Bottom = continuation(br, 1, setproperties(opts_br; ds = -opts_br.ds, max_steps = 14))

scene = plot(br, br1Top, br1Bottom; branchlabel = ["br", "br1Top", "br1Bottom"], legend = :topleft)

Algorithms

for the pitchfork bifurcation, the normal form is computed and non-trivial zeros are used to produce guesses for points on the bifurcated branch.

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

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:

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 Trapezoidal rule, you can pass PeriodicOrbitTrapProblem(M = 51) where M is the number of times slices in the periodic orbit.
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.
For Periodic orbits based on the shooting method, you need more parameters. For example, you can pass ShootingProblem(M, odeprob, Euler()) or PoincareShootingProblem(M, odeprob, Euler()) where odeprob::ODEProblem (see DifferentialEquations.jl) is an ODE problem to specify the Cauchy problem amd M is the number of sections.

Several examples are provided in 1d Brusselator (automatic) or 2d Ginzburg-Landau equation (finite differences, codim 2, Hopf aBS).

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

From Branch / Period-doubling point of curve of periodic orbits

We do not provide (for now) the automatic branching procedure for these bifurcations of periodic orbits. As a consequence, the user is asked to provide the amplitude of the bifurcated solution.

We provide the branching method for the following methods to compute periodic orbits: PeriodicOrbitTrapProblem,ShootingProblem and PoincareShootingProblem. The call is as follows. Please note that a deflation is included in this method to simplify branch switching.

An example of use is provided in Period doubling in Lur'e problem (PD aBS).

continuation(br::AbstractBranchResult, ind_PD::Int, contParams::ContinuationPar;
	δp = 0.1, ampfactor = 1, usedeflation = false, kwargs...)

From Bogdanov-Takens (BT) point to Fold / Hopf curve

We provide an automatic branch switching method in this case (see for example Extended Lorenz-84 model (codim 2 + BT/ZH aBS) or 2d Ginzburg-Landau equation (finite differences, codim 2, Hopf aBS)). Hence, you can perform automatic branch switching by calling continuation with the following options:

continuation(br::ContResult, ind_BT::Int,
	options_cont::ContinuationPar = br.contparams;
	nev = options_cont.nev,
	detect_codim2_bifurcation::Int = 0,
	start_with_eigen = false,
	autodiff = false,
	Teigvec = getvectortype(br),
	scaleζ = norm,
	kwargs...)

where ind_BT is the index of the BT point in br. Note that the BT has been detected during Fold or Hopf continuation. Calling the above method thus switches from Fold continuation to Hopf continuation (and vice-versa) 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.

From Zero-Hopf (ZH) point to Fold / Hopf curve

continuation(br::ContResult, ind_ZH::Int,
	options_cont::ContinuationPar = br.contparams;
	nev = options_cont.nev,
	detect_codim2_bifurcation::Int = 0,
	start_with_eigen = false,
	autodiff = false,
	Teigvec = getvectortype(br),
	scaleζ = norm,
	kwargs...)

where ind_ZH is the index of the ZH point in br. Note that the ZH has been detected during Fold or Hopf continuation. Calling the above method thus switches from Fold continuation to Hopf continuation (and vice-versa) 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.

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

continuation(br::ContResult, ind_HH::Int,
	options_cont::ContinuationPar = br.contparams;
	δp = nothing, ampfactor::Real = 1,
	nev = options_cont.nev,
	detect_codim2_bifurcation::Int = 0,
	start_with_eigen = false,
	autodiff = false,
	Teigvec = getvectortype(br),
	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.

From Bautin point to curve Folds of period orbits

From the Bautin normal form, we know that there is a curve of Fold of periodic orbits which branches from it.

We provide an automatic branch switching method in this case which reads as follows:

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

where prob is a method to compute periodic orbits (see From Hopf point to periodic orbits for more information).

Note that the two parameters in br will be used for the continuation of Fold points of periodic orbits.

From Zero-Hopf (ZH) point to curve NS of period orbits

From the Zero-Hopf normal form, we know that there is a curve of Neimark-Sacker (NS) bifurcations of periodic orbits which branches from it.

We provide an automatic branch switching method in this case which reads as follows:

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

where prob is a method to compute periodic orbits (see From Hopf point to periodic orbits for more information).

Note that the two parameters in br will be used for the continuation of NS points of periodic orbits.

From Hopf-Hopf (HH) point to curve NS of period orbits

From the Hopf-Hopf normal form, we know that there are two curves of Neimark-Sacker (NS) bifurcations of periodic orbits which branches from it.

We provide an automatic branch switching method in this case which reads as follows:

continuation(br::TwoParamCont, ind_HH::Int, 
	_contParams::ContinuationPar,
    prob::AbstractPeriodicOrbitProblem ;
    δp = nothing, ampfactor = 1, 
    whichns = 1,
    kwargs...)

where prob is a method to compute periodic orbits (see From Hopf point to periodic orbits for more information). The option whichns which belongs to {1,2} controls which NS curve you want to compute.

Note that the two parameters in br will be used for the continuation of NS points of periodic orbits.