Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps

Abstract

Invariant tori play a fundamental role in the dynamics of symplectic and volume-preserving maps. Codimension-one tori are particularly important as they form barriers to transport. Such tori foliate the phase space of integrable, volume-preserving maps with one action and $d$ angles. For the area-preserving case, Greene's residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits. Even though KAM theory applies to the three-dimensional case, the robustness of tori in such systems is still poorly understood. We study a three-dimensional, reversible, volume-preserving analogue of Chirikov's standard map with one action and two angles. We investigate the preservation and destruction of tori under perturbation by computing the "residue" of nearby periodic orbits. We find tori with Diophantine rotation vectors in the "spiral mean" cubic algebraic field. The residue is used to generate the critical function of the map and find a candidate for the most robust torus.

arXiv commentlaTeX, 40 pages, 26 figures

References in corpus (8)

• A lecture on the classical KAM theorem
• Chaotic advection in three-dimensional unsteady incompressible laminar flow
• Multidimensional continued fractions, dynamical renormalization and KAM theory
• Quadratic Volume Preserving Maps
• Quadratic volume preserving maps
• Renormalization and destruction of $1/{\gamma }^2$ tori in the standard nontwist map
• The Destruction of Tori in Volume-Preserving Maps
• Resonances and Twist in Volume-Preserving Mappings

Cited by in corpus (14)

• Thirty Years of Turnstiles and Transport
• Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps
• Birkhoff Averages and the Breakdown of Invariant Tori in Volume-Preserving Maps
• Critical Invariant Circles in Asymmetric and Multiharmonic Generalized Standard Maps
• Efficient Computation of Invariant Tori in Volume-Preserving Maps
• Accelerator modes and anomalous diffusion in 3D volume-preserving maps
• Trapping of swimmers in a vortex lattice
• Barriers to Transport and Mixing in Volume-Preserving Maps with Nonzero Flux
• Partial barriers to chaotic transport in 4D symplectic maps
• Mesochronic classification of trajectories in incompressible 3D vector fields over finite times
• Power-law trapping in the volume-preserving Arnold-Beltrami-Childress map
• Breakdown of rotational tori in 2D and 4D conservative and dissipative standard maps
• Global transport in a nonautonomous standard map
• A new method to compute periodic orbits in general symplectic maps