Resonances and Twist in Volume-Preserving Mappings

TL;DR

This paper investigates how volume-preserving maps near resonant tori can be approximated by standard maps, revealing conditions under which twist occurs and distinguishing these from classical KAM nondegeneracy conditions.

Contribution

It introduces a reduction of volume-preserving maps near resonances to standard maps and clarifies the twist condition in this context, differing from traditional KAM theory.

Findings

Near resonant tori, maps reduce to standard forms

Twist occurs when the frequency map crosses the resonance transversely

The twist condition differs from classical KAM nondegeneracy

Abstract

The phase space of an integrable, volume-preserving map with one action and $d$ angles is foliated by a one-parameter family of $d$-dimensional invariant tori. Perturbations of such a system may lead to chaotic dynamics and transport. We show that near a rank-one, resonant torus these mappings can be reduced to volume-preserving "standard maps." These have twist only when the image of the frequency map crosses the resonance curve transversely. We show that these maps can be approximated---using averaging theory---by the usual area-preserving twist or nontwist standard maps. The twist condition appropriate for the volume-preserving setting is shown to be distinct from the nondegeneracy condition used in (volume-preserving) KAM theory.