The Destruction of Tori in Volume-Preserving Maps

Abstract

Invariant tori are prominent features of symplectic and volume preserving maps. From the point of view of chaotic transport the most relevant tori are those that are barriers, and thus have codimension one. For an $n$-dimensional volume-preserving map, such tori are prevalent when the map is nearly "integrable," in the sense of having one action and $n-1$ angle variables. As the map is perturbed, numerical studies show that the originally connected image of the frequency map acquires gaps due to resonances and domains of nonconvergence due to chaos. We present examples of a three-dimensional, generalized standard map for which there is a critical perturbation size, ${\epsilon }_c$, above which there are no tori. Numerical investigations to find the "last invariant torus" reveal some similarities to the behavior found by Greene near a critical invariant circle for area preserving maps: the crossing time through the newly destroyed torus appears to have a power law singularity at ${\epsilon }_c$, and the local phase space near the critical torus contains many high-order resonances.

laTeX, 16 figures

References in corpus (2)

• A lecture on the classical KAM theorem
• On a theorem due to Birkhoff

Cited by in corpus (10)

• Frontiers of chaotic advection
• 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
• Efficient Computation of Invariant Tori in Volume-Preserving Maps
• Critical Invariant Circles in Asymmetric and Multiharmonic Generalized Standard Maps
• Accelerator modes and anomalous diffusion in 3D volume-preserving maps
• Barriers to Transport and Mixing in Volume-Preserving Maps with Nonzero Flux
• Localized shear generates three-dimensional transport
• Power-law trapping in the volume-preserving Arnold-Beltrami-Childress map