Global effect of non-conservative perturbations on homoclinic orbits

TL;DR

This paper analyzes how non-conservative, time-dependent perturbations influence homoclinic orbits in Hamiltonian systems, providing explicit formulas for the perturbed scattering map and illustrating with a rotator-pendulum example.

Contribution

It introduces explicit formulas for the perturbed scattering map in action-angle coordinates under non-conservative perturbations, extending the understanding of homoclinic dynamics.

Findings

Explicit formulas for the perturbed scattering map are derived.

The formulas are expressed as convergent integrals in action-angle coordinates.

Application demonstrated on a perturbed rotator-pendulum system.

Abstract

We study the effect of time-dependent, non-conservative perturbations on the dynamics along homoclinic orbits to a normally hyperbolic invariant manifold. We assume that the unperturbed system is Hamiltonian, and the normally hyperbolic invariant manifold is parametrized via action-angle coordinates. The homoclinic excursions can be described via the scattering map, which gives the future asymptotic of an orbit as a function of the past asymptotic. We provide explicit formulas, in terms of convergent integrals, for the perturbed scattering map expressed in action-angle coordinates. We illustrate these formulas in the case of a perturbed rotator-pendulum system.