Homoclinic to hyperbolic saddle

Consider the ODE problem written

\[\frac{du}{dt}=F(u(t),p)\tag{E}\]

where $p$ are parameters. A homoclinic solution $u^*$ to a hyperbolic saddle $u^s(p)$ satisfies $\lim\limits_{t\to\pm\infty}u^*(t) = u^s$ and $u^*(p)$ is a hyperbolic saddle of (E).

We provide 2 methods for computing such homoclinic orbits

1. one (Collocation) based on orthogonal collocation to discretize the above problem (E), with adaptive mesh
2. one (Shooting) based on parallel standard shooting

General method

The general method amounts to solving a boundary value problem which is simplified here for the exposition

\[\left\{\begin{aligned} & \dot{u}(t)-2 T\cdot F(u(t), p)=0 \\ & F\left(u^s, p\right)=0 \\ & Q^{U^{\perp}, \mathrm{T}}\left(u(0)-u^s\right)=0, \\ & Q^{S^{\perp}, \mathrm{T}}\left(u(1)-u^s\right)=0 \\ & \left\|u(0)-u^s\right\|-\epsilon_0=0 \\ & \left\|u(1)-u^s\right\|-\epsilon_1=0 \\ \end{aligned}\right.\]

Basically, we truncate the homoclinic orbit on $[-T,T]$ and we impose that $u(-T),u(T)$ is close to $u^s$ and belong to the stable / unstable subspaces of $u^s$. There are thus at most 3 free parameters and $T,\epsilon_0,\epsilon_1$ and the user can either

• chose one as a free parameter, for example $T$
• chose two as a free parameters, for example $T,\epsilon_1$

Continuation

Please see the tutorials for examples. In a nutshell, you can compute homolinic orbits by setting up a HomoclinicHyperbolicProblemPBC or by branching from a Bogdanov-Takens point.

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. Under the hood, the detection of these bifurcations is done by using Event detection as explained in Event Handling.

We refer to ^[DeWitte] for a description of the bifurcations.

Inclination-Flip with respect to the Stable / Unstable manifold is not yet detected.

References