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.
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updated 12 July 07