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

Anatoly Neishtadt, Alexey Okunev

Published 2021-08-19Version 1

We study averaging method for time-periodic perturbations of one-frequency Hamiltonian systems such that solutions of the perturbed system cross separatrices of the unperturbed system. The Hamiltonian depends on a parameter that slowly changes for the perturbed system (so Hamiltonian systems with two and a half degree of freedom are included in our class). We prove that for most initial conditions (except a set of measure $O(\sqrt{\varepsilon} |\ln^5 \varepsilon|)$, here $\varepsilon$ is the small parameter) the evolution of slow variables is described by the averaged system with accuracy $O(\sqrt{\varepsilon} |\ln \varepsilon|)$ over time $\sim \varepsilon^{-1}$.