Normally Elliptic Singular Perturbations and Persistence of Homoclinic Orbits

TL;DR

This paper investigates the persistence of homoclinic orbits in complex dynamical systems with fast and slow variables, deriving limit systems and manifold structures to understand their stability and behavior under perturbations.

Contribution

It introduces a method to analyze normally elliptic singular perturbations, constructing invariant manifolds and proving the persistence of homoclinic solutions in such systems.

Findings

Derived and justified the limit system for slow variables.

Constructed invariant manifolds with leading order approximations.

Proved the persistence of homoclinic solutions under perturbations.

Abstract

We consider a dynamical system, possibly infinite dimensional or non-autonomous, with fast and slow time scales which is oscillatory with high frequencies in the fast directions. We first derive and justify the limit system of the slow variables. Assuming a steady state persists, we construct the stable, unstable, center-stable, center-unstable, and center manifolds of the steady state of a size of order O(1) and give their leading order approximations. Finally, using these tools, we study the persistence of homoclinic solutions in this type of normally elliptic singular perturbation problems.