Averaging and passage through resonances in two-frequency systems near separatrices

TL;DR

This paper provides asymptotic estimates that validate the averaging method for two-frequency Hamiltonian systems near separatrices, addressing challenges posed by resonances and passage through separatrices.

Contribution

It offers realistic asymptotic estimates that justify averaging in systems with simultaneous resonances and separatrix passage, including slow-fast Hamiltonian systems.

Findings

Averaging method justified near separatrices with resonances

Asymptotic estimates account for passage through separatrices

Results applicable to two-frequency perturbations of integrable systems

Abstract

The averaging method is a classical powerful tool in perturbation theory of dynamical systems. There are two major obstacles to applying the averaging method, resonances and separatrices. In this paper we obtain realistic asymptotic estimates that justify the use of averaging method in a generic situation where both these obstacles are present at the same time, passage through a separatrix for time-periodic perturbations of one-frequency Hamiltonian systems. As a general phenomenon, resonances accumulate at separatrices. The Hamiltonian depends on a parameter that slowly changes for the perturbed system (so slow-fast Hamiltonian systems with two and a half degrees of freedom are included in our class). Our results can also be applied to perturbations of generic two-frequency integrable systems near separatrices, as they can be reduced to periodic perturbations of one-frequency systems.