Beyond the Melnikov method: a computer assisted approach

TL;DR

This paper introduces a computer-assisted, geometric method to verify transversal intersections of invariant manifolds in perturbed dynamical systems, extending Melnikov's approach without requiring explicit formulas or small perturbation assumptions.

Contribution

It provides a new geometric proof for NHIMs and their invariant manifolds, enabling computer-assisted verification of intersections over explicit parameter ranges.

Findings

Establishes bounds on derivatives of invariant manifolds.

Verifies intersections without explicit homoclinic formulas.

Applicable to a range of perturbation parameters.

Abstract

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds (NHIMs). The method is based on a new geometric proof of the normally hyperbolic invariant manifold theorem, which establishes the existence of a NHIM, together with its associated invariant manifolds and bounds on their first and second derivatives. We do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are `small enough', as is the case in the classical Melnikov approach.