Intersections of Lagrangian submanifolds and the Mel'nikov 1-form

TL;DR

This paper clarifies the geometric interpretation of Mel'nikov's method for identifying heteroclinic points between hyperbolic periodic orbits, using Lagrangian submanifold intersections and addressing convergence issues of the Mel'nikov 1-form.

Contribution

It develops a general theory of intersections for Lagrangian submanifolds constrained within auxiliary families, and explains how zeros of the Mel'nikov 1-form detect heteroclinic orbits.

Findings

Explicit geometric interpretation of Mel'nikov's method.

Integral expression for the Mel'nikov 1-form, with discussion on convergence.

Framework for detecting heteroclinic points via Lagrangian intersections.

Abstract

We make explicit the geometric content of Mel'nikov's method for detecting heteroclinic points between transversally hyperbolic periodic orbits. After developing the general theory of intersections for pairs of family of Lagrangian submanifolds constrained to live in an auxiliary family of submanifolds, we explain how the heteroclinic orbits are detected by the zeros of the Mel'nikov 1 -form. This 1 -form admits an integral expression, which is non-convergent in general. Finally, we discuss different solutions to this convergence problem.