Resonances and Twist in Volume-Preserving Mappings

Details Resonances and Twist in Volume-Preserving Mappings Resonances and Twist in Volume-Preserving Mappings

2010-03-03

arxiv.org/abs/1003.0922v2

Nonlinear Sciences

Chaotic Dynamics

Many typos fixed and notation simplified. New $n^{th}$ order averaging theorem and volume-preserving variant. Numerical compar

Scientific paper

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.

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