Homoclinic connection

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

Homoclinic, Heteroclinic Connections and Intersections.

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 $V$ be an connected invariant manifold such that

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

Then $V$ is called a homoclinic connection.

Heteroclinic connection

It is a similar notion, but it refers to two fixed points, $p$ and $q$. The condition satisfied by $V$ is replaced with:

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

This notion is not symmetric with respect to $p$ and $q$.

Homoclinic and heteroclinic intersections

When the invariant manifolds $W^{s}(f,p)$ and $W^{u}(f,q)$, possibly with $p=q$, intersect but there is no homoclinic/heteroclinic connection, a different structure is formed by the two manifolds, sometimes referred to as the homoclinic/heteroclinic tangle. The figure has a conceptual drawing illustrating their complicated structure. The theoretical result supporting the drawing is the lambda-lemma.

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.

Significance

When a dynamical system is perturbed, a homoclinic connection splits. It becomes a disconnected invariant set. Near it, there will be a chaotic set called Smale's horseshoe. Thus, the existence of a homoclinic connection is thus considered as potentially leading to chaos.