Homoclinic connection
In dynamical systems, a branch of mathematics, a structure formed from the stable manifold and unstable manifold of a fixed point.
![](http://upload.wikimedia.org/wikipedia/en/thumb/7/7c/Connections.png/220px-Connections.png)
Definition for maps
Let be a map defined on a manifold , with a fixed point . Let and be the stable manifold and the unstable manifold of the fixed point , respectively. Let be an connected invariant manifold such that
Then is called a homoclinic connection.
Heteroclinic connection
It is a similar notion, but it refers to two fixed points, and . The condition satisfied by is replaced with:
This notion is not symmetric with respect to and .
Homoclinic and heteroclinic intersections
When the invariant manifolds and , possibly with , 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
- There is some variation in the definition across various publications;
- 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 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.