Resonances and Twist in Volume-Preserving Mappings

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.

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

References in corpus (8)

• On universality of algebraic decays in Hamiltonian systems
• Meanders and Reconnection-Collision Sequences in the Standard Nontwist Map
• Universality of algebraic laws in Hamiltonian systems
• Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations
• Resonance Zones and Lobe Volumes for Volume-Preserving Maps
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• Chaotic Advection and the Emergence of Tori in the Küppers-Lortz State
• Normal Forms for Symplectic Maps with Twist Singularities

Cited by in corpus (7)

• Universal Fractional Map and Cascade of Bifurcations Type Attractors
• Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps
• Birkhoff Averages and the Breakdown of Invariant Tori in Volume-Preserving Maps
• Diffusion and Drift in Volume-Preserving Maps
• Barriers to Transport and Mixing in Volume-Preserving Maps with Nonzero Flux
• Resonance--Assisted Tunneling in Deformed Optical Microdisks with a Mixed Phase Space
• Power-law trapping in the volume-preserving Arnold-Beltrami-Childress map