Stationary phase methods and the splitting of separatrices

TL;DR

This paper develops explicit formulas for the Melnikov function in Hamiltonian systems with rapidly oscillating perturbations using stationary phase methods, applicable even without explicit separatrix knowledge and in non-analytic cases.

Contribution

It introduces a novel stationary phase approach to compute Melnikov functions explicitly, extending applicability to non-analytic and low-regularity perturbations.

Findings

Explicit Melnikov function formulas derived

Applicable to non-analytic and low-regularity systems

Illustrated with charged particle motion in electromagnetic fields

Abstract

Using stationary phase methods, we provide an explicit formula for the Melnikov function of the one and a half degrees of freedom system given by a Hamiltonian system subject to a rapidly oscillating perturbation. Remarkably, the Melnikov function turns out to be computable without an explicit knowledge of the separatrix and in the case of non-analytic systems. This is related to a priori stable systems coupled with low regularity perturbations. It also applies to perturbations controlled by wave-type equations, so in particular we also illustrate this result with the motion of charged particles in a rapidly oscillating electromagnetic field. Quasiperiodic perturbations are discussed too.