Fold / Hopf Continuation

In this page, we explain how to perform continuation of Fold / Hopf points and detect the associated bifurcations.

List of detected codim 2 bifurcation points

Bifurcationsymbol used
Bogdanov-Takensbt
Bautingh
Cuspcusp
Zero-Hopfzh
Hopf-Hopfhh

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 docs in BifurcationKit for more information.

Detection of codim 2 bifurcation points

You can detect the following codim 2 bifurcation points by using the option 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 of Bogdanov-Takens (BT) is performed using the test function[Bindel] $\psi_{BT}(p) = \langle w(p),v(p)\rangle$
  • the detection of Zero-Hopf (ZH) is performed 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.

Hopf continuation

The continuation of Fold bifurcation points is based on solving the extended system for $(u^*, v, \omega)$

\[\begin{aligned} & 0=\mathbf F\left(u^*, u^*; p\right) \\ & 0=\Delta\left(u^*, p, \mathrm{i} \omega\right) v \\ & 0=v^{\mathrm{H}} v-1 \end{aligned}\]

where $\Delta(\lambda)\cdot v := \lambda v - d_1\mathbf F(u^*,u^*; p)\cdot v-d_2\mathbf F(u^*, u^*; p)\cdot(e^{\lambda\cdot}v)$.

Detection of codim 2 bifurcation points

You can detect the following codim 2 bifurcation points by using the option detect_codim2_bifurcation in the method continuation.

  • the detection of Bogdanov-Takens (BT) is performed using the test function $\psi_{BT}(p) = \omega$
  • the detection of Bautin (GH) is based on the test function $\psi_{GH}(p) = \Re(l_1(p))$ where $l_1$ is the Lyapunov coefficient defined in Simple Hopf point.
  • the detection of Zero-Hopf (ZH) is performed by monitoring the eigenvalues.
  • the detection of Hopf-Hopf (HH) is performed by monitoring the eigenvalues.

The continuation of Hopf points is stopped at BT and when $\omega<100\epsilon$ where $\epsilon$ is the newton tolerance.

Codim 2 continuation

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

Missing docstring.

Missing docstring for continuation(br::BifurcationKit.AbstractBranchResult, ind_bif::Int64, lens2::Lens, options_cont::ContinuationPar = br.contparams ; kwargs...). Check Documenter's build log for details.

where the options are as above except with have an additional parameter axis lens2 which is used to locate the bifurcation points.