Asymptotic Expansion of the Homoclinic Splitting Matrix for the Rapidly, Quasiperiodically, Forced Pendulum

TL;DR

This paper develops an asymptotic expansion for the homoclinic splitting matrix in a Hamiltonian system with a pendulum coupled to oscillators, revealing exponentially small contributions as hyperbolicity vanishes.

Contribution

It introduces a novel asymptotic expansion method for the homoclinic splitting matrix in a complex Hamiltonian system with quasiperiodic forcing.

Findings

Derived an exponentially small upper bound on the splitting matrix

Provided a shift-of-contour argument to analyze the asymptotic behavior

Enhanced understanding of homoclinic splitting in quasiperiodically forced systems

Abstract

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, devising an asymptotic expansion for the splitting (matrix) associated with a homoclinic point. This expansion consists of contributions that are manifestly exponentially small in the limit of vanishing hyperbolicity, by a shift-of-contour argument. Hence, we infer a similar upper bound on the splitting itself.