Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

TL;DR

This paper investigates the complex dynamics of a five-parameter quadratic volume-preserving map in three dimensions, focusing on invariant circles, bifurcations, and the creation of vortex-bubbles near saddle-center-Neimark-Sacker bifurcations.

Contribution

It introduces an algorithm to compute elliptic invariant circles and analyzes their bifurcations and resonance structures in a novel quadratic volume-preserving map.

Findings

Elliptic invariant circles are accurately computed and classified by resonance.

Rational rotation numbers lead to 'string of pearls' bifurcations.

Complex vortex-bubble structures form near saddle-center-Neimark-Sacker 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.