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.
Abdullaev
S S
SS
Institut für Plasmaphysik, Forschungszentrum Jülich GmbH, EURATOM Association, Trilateral Euregio Cluster, D-52425 Jülich, Germany.