Canonical maps near separatrix in Hamiltonian systems.

A systematic and rigorous method to construct symplectic maps near separatrix of generic Hamiltonian systems subjected to time-periodic perturbations is developed. It is based on the method of canonical transformation of variables to construct Hamiltonian maps [J. Phys. A 35, 2811 (2002)]]. Using canonical transformation of variables and the first-order approximation for the generating function, the general form of mapping in terms of time and energy variables is obtained. Different limiting cases of the mapping are considered. The method is illustrated for simple Hamiltonian systems with one and a large number of saddle points. It is also applied to derive mappings for the periodic-driven Morse oscillator describing the process of stochastic excitation and dissociation of diatomic molecules. The so-called canonical Kepler map is derived for the one-dimensional hydrogen atom in a microwave field.