Homoclinic connection

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{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In dynamical systems, a branch of mathematics, a structure formed from the stable manifold and unstable manifold of a fixed point.

File:Connections.png
Homoclinic, Heteroclinic Connections and Intersections.

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. Homoclinic tangles are always accompanied by a Smale horseshoe.

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 both forwards and backwards in time. A pendulum in the absence of friction is an example of a mechanical system that does have a homoclinic connection. When the pendulum is released from the top position (the point of highest potential energy), with infinitesimally small velocity, the pendulum will return to the same position. Upon return, it will have exactly the same velocity. The time it will take to return will increase to as the initial velocity goes to zero. One of the demonstrations in the pendulum article exhibits this behavior.

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 can potentially lead to chaos. For example, when a pendulum is placed in a box, and the box is subjected to small horizontal oscillations, the pendulum may exhibit chaotic behavior.

See also