Homoclinic structures in the standard map
You see below the standard map orbits at the saddle point (0,0).
Entangled orbits near separatrises correspond to chaotic dynamics
(we should find positive Lyapunov exponents but in a numerical experiment
we can't exclude very very long chaotic transient too)
In the second picture you see stable and unstable separatrises
of the saddle point (unfortunately two pictures have different centering).
Therefore there are countable set of periodic and continuum of chaotic
orbits. Unfortunately we don't know if the measure of this region is non zero
and why these separatrises are so winding.
Previous: The Standard map
updated 12 July 07