Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in (mathbb{R}^4) with (mathbb{Z}_2)-symmetry and integral of motion

TL;DR

This paper studies the invariant manifolds of homoclinic orbits in a symmetric four-dimensional flow with an integral of motion, showing that super-homoclinics lead to complex dynamics including infinitely many multi-pulse homoclinic loops.

Contribution

It provides criteria for the existence of invariant manifolds and demonstrates how super-homoclinics induce rich dynamical behavior in symmetric systems.

Findings

Existence criteria for stable and unstable manifolds of homoclinic orbits.

Transverse intersections of these manifolds create super-homoclinics.

Presence of super-homoclinics implies infinitely many multi-pulse homoclinic loops.

Abstract

We consider a (mathbb{Z}_2)-equivariant flow in (mathbb{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 Schr\"odinger equations is considered.