Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

Abstract

We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R^3. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and also is the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with intersecting two-dimensional stable and unstable manifolds that bound a spherical ``vortex-bubble''. We show that this occurs near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at least in its normal form, an elliptic invariant circle. We develop a simple algorithm to accurately compute these elliptic invariant circles and their longitudinal and transverse rotation numbers and use it to study their bifurcations, classifying them by the resonances between the rotation numbers. In particular, rational values of the longitudinal rotation number are shown to give rise to a string of pearls that creates multiple copies of the original spherical structure for an iterate of the map.

53 pages, 29 figures

References in corpus (3)

• Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation
• Variational method for locating invariant tori
• Nilpotent normal form for divergence-free vector fields and volume-preserving maps

Cited by in corpus (10)

• Global structure of regular tori in a generic 4D symplectic map
• Bifurcations of families of 1D-tori in 4D symplectic maps
• Accelerator modes and anomalous diffusion in 3D volume-preserving maps
• Resonances and Twist in Volume-Preserving Mappings
• Resonant Tori, Transport Barriers, and Chaos in a Vector Field with a Neimark-Sacker Bifurcation
• Dynamics of the Chaplygin ball on a rotating plane
• Visualizing Attractors of the Three-Dimensional Generalized Hénon Map
• Interpolating vector fields for near identity maps and averaging
• Power-law trapping in the volume-preserving Arnold-Beltrami-Childress map
• Topological Dynamics of Volume-Preserving Maps Without an Equatorial Heteroclinic Curve