Non-conservative perturbations of homoclinic snaking scenarios

Abstract

Homoclinic snaking refers to the continuation of homoclinic orbits to an equilibrium E near a heteroclinic cycle connecting E and a periodic orbit P. Typically homoclinic snaking appears in one-parameter families of reversible, conservative systems.Here we discuss perturbations of this scenario which are both non-reversible and non-conservative. We treat this problem analytically in the spirit of the work [3]. The continuation of homoclinic orbits happens with respect to both the original continuation parameter μ and the perturbation parameter λ. The continuation curves are parametrised by the dwelling time L of the homoclinic orbit near P. It turns out that λ(L) tends to zero while the μ vs. L diagram displays isolas or criss-cross snaking curves in a neighbourhood of the original snakes-and-ladder structure.In the course of our studies we adapt both Fenichel coordinates near P and the analysis of Shilnikov problems near P to the present situation.