Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in R4 with Z2-symmetry and integral of motion

Abstract

We consider a Z(2)-equivariant flow in R-4 with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Gamma. We provide criteria for the existence of stable and unstable invariant manifolds of Gamma. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrodinger equations is considered. (C) 2022 The Authors. Published by Elsevier Inc.