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

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

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