Averaging and Passage Through Resonances in Two-Frequency Systems Near Separatrices

In this paper we obtain sharp asymptotic estimates for the accuracy of the averaging method for time-periodic perturbations of one-frequency Hamiltonian systems while passing through a separatrix. The Hamiltonian depends on a parameter that slowly changes for the perturbed system (thus, slow–fast Hamiltonian systems with two and a half degrees of freedom are included in our class). Let \(\varepsilon \) be the small parameter of the system, then under certain genericity conditions we prove that the accuracy of averaging is \(O(\sqrt{\varepsilon }|\ln \varepsilon |)\) for times of order \(\varepsilon ^{-1}\) (such times correspond to a change of slow variables of order 1) for all initial data outside an exceptional set with the measure \(O(\sqrt{\varepsilon }|\ln ^5 \varepsilon |)\). The main novelty of the paper lies in estimating the scattering amplitude and the measure of captured orbits while passing through resonances near separatrices. 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.