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.
