Homoclinic connection: Difference between revisions
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Let <math>f:M\to M</math> be a [[map]] defined on a manifold <math>M</math>, with a fixed point <math>p</math>. |
Let <math>f:M\to M</math> be a [[map]] defined on a manifold <math>M</math>, with a fixed point <math>p</math>. |
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Let <math>W^s(f,p)</math> and <math>W^u(f,p)</math> be the [[stable manifold]] and the [[unstable manifold]] |
Let <math>W^s(f,p)</math> and <math>W^u(f,p)</math> be the [[stable manifold]] and the [[unstable manifold]] |
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of the fixed point <math>P</math>, respectively. Let <math> |
of the fixed point <math>P</math>, respectively. Let <math>V</math> be an [[connected space |connected]] [[invariant manifold]] such that |
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:<math> V\subseteq W^s(f,p)\cup W^u(f,p)</math> |
:<math> V\subseteq W^s(f,p)\cup W^u(f,p)</math> |
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Then <math>V</math> is called a '''homoclinic connection'''. |
Then <math>V</math> is called a '''homoclinic connection'''. |
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== Heteroclinic connection == |
== Heteroclinic connection == |
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It is a similar notion, but it refers to two fixed points, <math>p</math> and <math>q</math>. The condition satisfied by |
It is a similar notion, but it refers to two fixed points, <math>p</math> and <math>q</math>. The condition satisfied by |
Revision as of 21:19, 26 November 2010
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.