Homoclinic Tangle

Refer to the above figures. Let  be the point of intersection, with  ahead of  on one manifold and  ahead of  of the other. The mapping of each of these points  and  must be ahead of the mapping of , . 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  another homoclinic point. Since  is closer to the hyperbolic point than , the distance between  and  is less than that between  and . 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.

The homoclinic tangle is the same topological structure as the Smale horseshoe map.

REFERENCES:

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 145, 1989.