Resonance Zones and Lobe Volumes for Volume-Preserving 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.

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References in corpus (4)

• Heteroclinic intersections between Invariant Circles of Volume-Preserving Maps
• Canonical Melnikov theory for diffeomorphisms
• A variational principle for volume-preserving dynamics
• Generating Forms for Exact Volume-Preserving Maps

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• Geometric determination of heteroclinic and unstable periodic orbit classical actions