Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in [formula omitted] with [formula omitted]-symmetry and integral of motion

Abstract

We consider a Z2-equivariant flow in R4 with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. 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 Schrödinger equations is considered.