Canonical Melnikov theory for diffeomorphisms

TL;DR

This paper develops a canonical first-order deformation calculus for invariant manifolds in perturbed diffeomorphisms, generalizing classical Melnikov theory to detect transverse intersections of stable and unstable manifolds.

Contribution

It introduces a canonical form of the Melnikov function as a section of the normal bundle, unifying and extending classical methods for various types of maps.

Findings

Generalized Melnikov function as a normal bundle section

Reproduces classical Melnikov and Poincaré methods

Detects transverse intersections via zeros of the Melnikov displacement

Abstract

We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection. We show how our definition reproduces the classical methods of Poincar\'{e} and Melnikov and specializes to methods previously used for exact symplectic and volume-preserving maps. We use the method to detect the transverse intersection of stable and unstable manifolds and relate this intersection to the set of zeros of the Melnikov displacement.