From codim 2 (equilibria) to periodic orbits
From Bautin point to curve Folds of periodic orbits
From the Bautin normal form, we know that there is a curve of Fold of periodic orbits near the bifurcation point.
We provide an automatic branch switching method in this case which reads as follows:
continuation(br::HopfCont, ind_BAUTIN::Int, _contParams::ContinuationPar, disc::AbstractBoundaryValueDiscretization ; δp = nothing, ampfactor::Real = 1, detect_codim2_bifurcation::Int = 0, autodiff = true, kwargs...)where disc is a method to discretize periodic orbits, either Periodic orbits based on orthogonal collocation or Periodic orbits based on the shooting method (see From Hopf point to periodic orbits for more information). Note that the Bautin point must have been detected during Hopf point continuation.
Note that the two parameters in br will be used for the continuation of Fold points of periodic orbits.
See Lorenz-84 model, take 2 for an example of use.
From Zero-Hopf (ZH) point to curve NS of periodic orbits
From the Zero-Hopf normal form, we know that there is a curve of Neimark-Sacker (NS) bifurcations of periodic orbits near the bifurcation point.
We provide an automatic branch switching method in this case which reads as follows:
continuation(br::TwoParamCont, ind_ZH::Int, _contParams::ContinuationPar, disc::AbstractBoundaryValueDiscretization ; δp = nothing, ampfactor::Real = 1, detect_codim2_bifurcation::Int = 0, autodiff = true, kwargs...)where disc is a method to discretize periodic orbits, either Periodic orbits based on orthogonal collocation or Periodic orbits based on the shooting method (see From Hopf point to periodic orbits for more information). Note that the ZH point can have been detected during Fold or Hopf point continuation.
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 periodic orbits
From the Hopf-Hopf normal form, we know that there are two curves of Neimark-Sacker (NS) bifurcations of periodic orbits near the bifurcation point.
We provide an automatic branch switching method in this case which reads as follows:
continuation(br::TwoParamCont, ind_HH::Int, _contParams::ContinuationPar, disc::AbstractBoundaryValueDiscretization ; δp = nothing, ampfactor::Real = 1, whichns = 1, detect_codim2_bifurcation::Int = 0, autodiff = true, kwargs...)where disc is a method to discretize periodic orbits, either Periodic orbits based on orthogonal collocation or Periodic orbits based on the shooting method (see From Hopf point to periodic orbits for more information). Note that the HH point must have been detected during Hopf point continuation. 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.
See Lorenz-84 model, take 2 for an example of use.