Nonreversible homoclinic snaking

Abstract

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 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. Keywords: global bifurcationhomoclinic snakingheteroclinic cycleAMS Subject Classifications: 37C2737C2937G2037G25 Acknowledgements M. Vielitz was supported by the German National Academic Foundation (Studienstiftung des deutschen Volkes). The authors are grateful to V. Kirk for pointing out related results.