Continuation of Fold of periodic orbits
In this page, we explain how to perform continuation of Fold points and detect the associated codim 2 bifurcations.
List of detected codim 2 bifurcation points
Bifurcation | symbol used | Multipliers |
---|---|---|
Cusp | cusp | {1,1} |
Strong resonance 1:1 bifurcation | R1 | {1,1,1} |
Fold / Flip | foldPD | {1,1,-1} |
Fold / Neimark-Sacker | foldNS | {1,1,$e^{\pm i\alpha }$} |
In a nutshell, all you have to do (see below) is to call continuation(br, ind_bif, lens2)
to continue the bifurcation point stored in br.specialpoint[ind_bif]
and set proper options.
Fold continuation
The continuation of Fold bifurcation points is based on a Minimally Augmented[Govaerts] formulation which is an efficient way to detect singularities (see Fold / Hopf Continuation). All the methods (see Periodic orbits computation) for computing periodic orbits are compatible with this algorithm. In particular, you can perform these computations in large dimensions.
Detection of codim 2 bifurcation points
You can detect the following codim 2 bifurcation points by using the keyword argument detect_codim2_bifurcation
in the method continuation
- the detection of Cusp (Cusp) is done by the detection of Fold bifurcation points along the curve of Folds by monitoring the parameter component of the tangent.
- the detection the above bifurcation points is done by monitoring the number of eigenvalues $\lambda$ such that $\Re\lambda > \min\limits_{\nu\in\Sigma(dF)}|\Re\nu|$ and $\Im\lambda > \epsilon$ where $\epsilon$ is the Newton tolerance.
Setting the jacobian
In order to apply the newton algorithm to the Fold functional, one needs to invert the jacobian. This is not completely trivial as one must compute this jacobian and then invert it. You can select the following jacobians for your computations (see below):
- [Default] for
jacobian_ma = :autodiff
, automatic differentiation is applied to the Fold functional and the matrix is then inverted using the provided linear solver. In particular, the jacobian is formed. This is very well suited for small dimensions (say < 100) - for
jacobian_ma = :finiteDifferences
, same asjacobian_ma = :autodiff
but the jacobian is computed using finite differences. - for
jacobian_ma = :minaug
, a specific procedure for evaluating the jacobian and inverting it (without forming the jacobian!) is used. This is well suited for large dimensions.
Codim 2 continuation
To compute the codim 2 curve of Fold points of periodic orbits, one can call continuation
with the following options
BifurcationKit.continuation
— Functioncontinuation(br, ind_bif, lens2; ...)
continuation(
br,
ind_bif,
lens2,
options_cont;
prob,
start_with_eigen,
detect_codim2_bifurcation,
update_minaug_every_step,
kwargs...
)
Codimension 2 continuation of Fold / Hopf points. This function turns an initial guess for a Fold / Hopf point into a curve of Fold / Hopf points based on a Minimally Augmented formulation. The arguments are as follows
br
results returned after a call to continuationind_bif
bifurcation index inbr
lens2
second parameter used for the continuation, the first one is the one used to computebr
, e.g.getlens(br)
options_cont = br.contparams
arguments to be passed to the regular continuation
Optional arguments:
bdlinsolver
bordered linear solver for the constraint equationupdate_minaug_every_step
update vectorsa, b
in Minimally Formulation everyupdate_minaug_every_step
stepsstart_with_eigen = false
whether to start the Minimally Augmented problem with information from eigen elementsdetect_codim2_bifurcation ∈ {0,1,2}
whether to detect Bogdanov-Takens, Bautin and Cusp. If equals1
non precise detection is used. If equals2
, a bisection method is used to locate the bifurcations.kwargs
keywords arguments to be passed to the regular continuation
where the parameters are as above except that you have to pass the branch br
from the result of a call to continuation
with detection of bifurcations enabled and index
is the index of Hopf point in br
you want to refine.
For ODE problems, it is more efficient to use the Matrix based Bordered Linear Solver passing the option bdlinsolver = MatrixBLS()
It is recommended that you use the option start_with_eigen = true
where br
is a branch of periodic orbits and the options are as above except with have an additional parameter axis lens2
which is used to locate the bifurcation points.
References
- Govaerts
Govaerts, Willy J. F. Numerical Methods for Bifurcations of Dynamical Equilibria. Philadelphia, Pa: Society for Industrial and Applied Mathematics, 2000.