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.
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updated 12 July 2007