Homoclinic connection

In dynamical systems, a branch of mathematics, a structure formed from the stable manifold and unstable manifold of a fixed point.

Definition for maps

Let $f:M\to M$ be a map defined on a manifold $M$, with a fixed point $p$. Let $W^{s}(f,p)$ and $W^{u}(f,p)$ be the stable manifold and the unstable manifold of the fixed point $P$, respectively. Let $U$ be an connected invariant manifold such that

$U\subseteq W^{s}(f,p)\cup W^{u}(f,p)$

Then $U$ is called a homoclinic connection.

Definition for continuous flows

For continuous flows, the definition is essentially the same.

Comments

1. There is some variation in the definition across various publications;
2. Historically, the first case considered was that of a continuous flow on the plane, induced by an ordinary differential equation. In this case, a homoclinic connection is a single trajectory that converges to the fixed point $p$ both forwards and backwards in time.