Refer to the above figures. 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 T X' and T X'' must be ahead of the mapping of X, T X. 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^2 X another homoclinic point. Since T^2 X is closer to the hyperbolic point than T X, the distance between T^2 X and T X is less than that between X and T X. 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.

homoclinic point | Smale horseshoe map