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In [[dynamical systems]], a branch of [[mathematics]], a structure formed from the [[stable manifold]] and [[unstable manifold]] of a [[Fixed point (mathematics)|fixed point]].  
[[File:Connections.png|thumb|Homoclinic, Heteroclinic Connections and Intersections.]]
 
== Definition for maps ==
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>W^s(f,p)</math> and <math>W^u(f,p)</math> be the [[stable manifold]] and the [[unstable manifold]]
of the fixed point <math>P</math>, respectively. Let <math>V</math> be an [[connected space|connected]] [[invariant manifold]] such that
:<math> V\subseteq W^s(f,p)\cup W^u(f,p)</math>
Then <math>V</math> is called a '''homoclinic connection'''.
 
== Heteroclinic connection ==
It is a similar notion, but it refers to two fixed points, <math>p</math> and <math>q</math>. The condition satisfied by
<math>V</math> is replaced with:
:<math>V\subseteq W^s(f,p)\cup W^u(f,q)</math>
This notion is '''not symmetric''' with respect to <math>p</math> and <math>q</math>.
 
== Homoclinic and heteroclinic intersections ==
When the invariant manifolds <math>W^s(f,p)</math> and <math>W^u(f,q)</math>, possibly with <math>p=q</math>, 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 [[Inclination lemma|lambda-lemma]]. Homoclinic tangles are always accompanied by a [[Smale horseshoe]].
 
== Definition for continuous flows ==
For continuous [[flow (mathematics)|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 (geometry)|plane]], induced by an [[ordinary differential equation]]. In this case, a homoclinic connection is a single [[trajectory]] that converges to the fixed point <math>p</math> both forwards and backwards in time. A [[pendulum (mathematics)|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 <math>\infty</math> as the initial velocity goes to zero. One of the demonstrations in the [[pendulum (mathematics)|pendulum]] article exhibits this behavior.
 
== Significance ==
When a dynamical system is perturbed, a homoclinic connection '''splits'''. It becomes a [[connected space|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 theory|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==
* [[Homoclinic orbit]]
* [[Heteroclinic orbit]]
 
[[Category:Dynamical systems]]

Latest revision as of 22:40, 1 September 2014

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