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 crossings of stable and unstable separatrises
of the saddle point (unfortunately two pictures have different centering).
Therefore there are Smale horseshoe, countable set of periodic
and continuum of chaotic orbits.
Previous: The Standard map
updated 12 July 2007