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

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):

• jacobian_ma = :autodiff [Default]: 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)
• jacobian_ma = :finiteDifferences: same as jacobian_ma = :autodiff but the jacobian is computed using finite differences.
• 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 and for matrix-free version.
• jacobian_ma = :MinAugMatrixBased the jacobian matrix is evaluated using analytical formula. This is faster than :autodiff.

Codim 2 continuation

To compute the codim 2 curve of Fold points of periodic orbits, one can call continuation with the following options

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