Analysis of the separatrix map in Hamiltonian systems

A one-degree-of-freedom Hamiltonian system with time periodic perturbation generally display rich dynamics near the separatrix, the understanding of which is important in studying transport and diffusion. We introduce the exact separatrix map which provides an effective way to describe dynamics near the separatrix. We show that the separatrix map and its inverse satisfy twist conditions with logarithmic singularities and, therefore, they have an infinite twist near the transversal intersections of the stable and unstable manifolds of the hyperbolic equilibrium. Using the variational formalism with the generating functions, we show the dynamics near the singularities correspond to the anti-integrable limit. As a consequence, we show the nonexistence of rotational invariant circles with sufficiently large rotation number, first obtained by Lazutkin in standard maps. Moreover, the existence of a uniformly hyperbolic chaotic invariant set is shown in connection with the theory of the anti-integrable limit. The simplest form of the separatrix map is studied in detail to illustrate some of the above results explicitly.