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Homoclinic Point


A point where a stable and an unstable separatrix (invariant manifold) from the same fixed point or same family intersect. Therefore, the limits

 lim_(k->infty)f^k(X)

and

 lim_(k->-infty)f^k(X)

exist and are equal.

A small disk centered near a homoclinic point includes infinitely many periodic points of different periods. Poincaré showed that if there is a single homoclinic point, there are an infinite number. More specifically, there are infinitely many homoclinic points in each small disk (Nusse and Yorke 1996).


See also

Heteroclinic Point, Homoclinic Tangle, Manifold, Separatrix

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References

Nusse, H. E. and Yorke, J. A. "Basins of Attraction." Science 271, 1376-1380, 1996.Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 145, 1989.

Referenced on Wolfram|Alpha

Homoclinic Point

Cite this as:

Weisstein, Eric W. "Homoclinic Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HomoclinicPoint.html

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