Non-reversible perturbations of homoclinic snaking scenarios

Abstract

Homoclinic snaking refers to the continuation curves of homoclinic orbits near a heteroclinic cycle, which connects an equilibrium and a periodic orbit in a reversible Hamiltonian system. We consider non-reversible perturbations of this situation and show analytically that such perturbations typically lead to either infinitely many closed continuation curves (isolas) or to two snaking continuation curves, which follow the primary sinusoidal continuation curves alternately (criss-cross snaking). These two scenarios are illustrated numerically with computations for perturbed versions of the Swift–Hohenberg equation.