Beyond the Melnikov method II: multidimensional setting

TL;DR

This paper introduces a computer-assisted Melnikov-type method for verifying transversal intersections of invariant manifolds in multidimensional systems, applicable over explicit parameter ranges without requiring explicit homoclinic formulas.

Contribution

It extends the Melnikov method to multidimensional settings, eliminating the need for explicit homoclinic formulas and smallness assumptions, using rigorous numerics for parameter range verification.

Findings

Establishes transversal intersections without explicit homoclinic formulas

Applicable to a range of parameters, not just small perturbations

Uses rigorous numerics for bounds and verification

Abstract

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds. 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.