Homoclinic connection
Appearance
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 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 .
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.