Homoclinic to hyperbolic saddle

Consider the ODE problem written

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

where $p$ denotes the 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. Homoclinic based on orthogonal collocation: orthogonal collocation to discretize the above problem (E), with adaptive mesh,
2. Homoclinic based on parallel multiple shooting: standard shooting based on the flow of (E).

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 $u(-T)$ and $u(T)$ are close to $u^s$ and belong to the stable / unstable subspaces of $u^s$.

The homoclinic solution is thus parametrized by the three scalars $T$, $\epsilon_0$ and $\epsilon_1$. Besides the continuation parameter, the user must select the free parameters of the problem (keyword freeparams in HomoclinicHyperbolicProblemPBC); at most two of $T, \epsilon_0, \epsilon_1$ can be free, e.g.

• one free parameter, for example $T$,
• two free parameters, for example $T,\epsilon_1$.

Continuation

The homoclinic problem can be set up with generate_hom_problem (from a periodic orbit) or by branch switching from a Bogdanov–Takens point with continuation; see the tutorials for examples.

Detection of codimension 2 bifurcation points

Codimension-two bifurcation points along the homoclinic branch can be detected during the continuation. We refer to the page Detection of bifurcation points for the list of detected bifurcations and how to enable them.

References