On the phase change for perturbations of Hamiltonian systems with separatrix crossing

TL;DR

This paper analyzes phase changes in perturbed Hamiltonian systems near separatrices, providing estimates for averaged systems and unifying different probability definitions of capture into domains after crossing.

Contribution

It introduces a second-order averaged system approach to estimate phase change and unifies probability definitions of capture in separatrix crossing.

Findings

Derived a formula for phase change near separatrices for general perturbations.

Proved the accuracy of the second-order averaged system near separatrices.

Showed that different probability definitions yield the same capture probability.

Abstract

We study the evolution of angular variable (phase) for general (not necessarily Hamiltonian) perturbations of Hamiltonian systems with one degree of freedom near separatrices of the unperturbed system. To this end, we use averaged system of order 2. We obtain estimates for the accuracy of order 2 averaged system near separatrices and use these estimates to prove a formula for the phase change when solutions of the perturbed system approach separatrices of the unperturbed system (such formula is known when the perturbation is Hamiltonian). As an application of this formula, we show that two natural definitions of probability of capture into different domains after separatrix crossing proposed by V.I. Arnold and D.V. Anosov lead to the same formula for this probability.