Resonance Zones and Lobe Volumes for Volume-Preserving Maps

TL;DR

This paper develops a theoretical framework for analyzing resonance zones and lobe volumes in volume-preserving maps, connecting heteroclinic connections, generating forms, and Melnikov functions to understand phase space transport.

Contribution

It introduces a general theory of lobes for volume-preserving diffeomorphisms, linking lobe volume to integrals over heteroclinic intersections and extending classical action formulas.

Findings

Lobe volume is given by an integral of a generating form over heteroclinic intersections.

In perturbations, lobe volume reduces to an integral of a Melnikov function.

The theory generalizes classical results from planar twist maps.

Abstract

We study exact, volume-preserving diffeomorphisms that have heteroclinic connections between a pair of normally hyperbolic invariant manifolds. We develop a general theory of lobes, showing that the lobe volume is given by an integral of a generating form over the primary intersection, a subset of the heteroclinic orbits. Our definition reproduces the classical action formula in the planar, twist map case. For perturbations from a heteroclinic connection, the lobe volume is shown to reduce, to lowest order, to a suitable integral of a Melnikov function.