Nonreversible Homoclinic Snaking

TL;DR

This paper rigorously analyzes homoclinic snaking phenomena in non-reversible systems, revealing how manifold interactions and Floquet multipliers influence the existence and structure of homoclinic orbits, supported by numerical verification.

Contribution

It provides the first analytical verification of homoclinic snaking in non-reversible systems and explores the dependence on Floquet multipliers, including a nonsnaking scenario.

Findings

Homoclinic snaking occurs under specific manifold conditions.

The sign of Floquet multipliers affects snaking behavior.

Numerical example confirms theoretical assumptions.

Abstract

Homoclinic snaking refers to the sinusoidal snaking continuation curve of homoclinic orbits near a heteroclinic cycle connecting an equilibrium E and a periodic orbit P. Along this curve the homoclinic orbit performs more and more windings about the periodic orbit. Typically this behaviour appears in reversible Hamiltonian systems. Here we discuss this phenomenon in systems without any particular structure. We give a rigorous analytical verification of homoclinic snaking under certain assumptions on the behaviour of the stable and unstable manifolds of E and P. We show how the snaking behaviour depends on the signs of the Floquet multipliers of P. Further we present a nonsnaking scenario. Finally we show numerically that these assumptions are fulfilled in a model equation.