Abstract
We consider an a priori unstable (initially hyperbolic) near-integrable Hamiltonian system in a neighborhood of stable and unstable asymptotic manifolds of a family of hyperbolic tori. Such a neighborhood contains the most chaotic part of the dynamics. The main result of the paper is the construction of the separatrix map as a convenient tool for the studying of such dynamics. We present evidence that the separatrix map combined with the method of anti-integrable limit can give a large class of chaotic trajectories as well as diffusion trajectories.
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