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

$$\lim_{k\to\infty} f^k(X)$$

and

$$\lim_{k\to -\infty} f^k(X)$$

exist and are equal.

Refer to the above figure. Let $X$ be the point of intersection, with $X'$ ahead of $X$ on one Manifold and $X''$ ahead of $X$ of the other. The mapping of each of these points $TX'$ and $TX''$ must be ahead of the mapping of $X$, $TX$. The only way this can happen is if the Manifold loops back and crosses itself at a new homoclinic point. Another loop must be formed, with $T^2X$ another homoclinic point. Since $T^2X$ is closer to the hyperbolic point than $TX$, the distance between $T^2X$ and $TX$ is less than that between $X$ and $TX$. Area preservation requires the Area to remain the same, so each new curve (which is closer than the previous one) must extend further. In effect, the loops become longer and thinner. The network of curves leading to a dense Area of homoclinic points is known as a homoclinic tangle or tendril. Homoclinic points appear where Chaotic regions touch in a hyperbolic Fixed Point.

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, Manifold, Separatrix

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.