Asymptotical forms of canonical mappings near separatrix of Hamiltonian systems

Asymptotical behavior of canonical mappings near the separatrix of Hamiltonian systems subjected to time-periodic perturbations is studied. Based on general forms of these mappings [S. S. Abdullaev, Phys. Rev. E 70, 046202 (2004)] it is shown that the Melnikov-type integrals determining their generating functions can be presented as a sum of regular, R-(reg)(h), and oscillatory, R-(osc)(h), parts. General asymptotical formulas for R-(osc)(h) are derived. The oscillatory parts have zeros at primary resonant values of energy. Conditions are found at which the oscillatory parts, R-(osc)(h), can be neglected in the generating functions thus allowing us to obtain simplified mappings depending only the regular parts R-(reg)(h). Since the latter are smooth functions of energy h this allows us also to justify the widely used conventional separatrix mapping determined by R-(reg)(h) at the separatrix h=0. A theory is illustrated for a specific example of a Hamiltonian system, a particle dynamics in periodically perturbed double-well potential.