Peeling Bifurcations of Toroidal Chaotic Attractors

TL;DR

This paper studies the complex bifurcation phenomena of toroidal chaotic attractors, revealing how their topology and symmetry properties change through peeling bifurcations using advanced topological and symbolic methods.

Contribution

It introduces a novel analysis of double covers of toroidal attractors, combining kneading theory and circle maps to understand their bifurcations and topological transformations.

Findings

Identification of morphological changes in attractors during peeling bifurcations

Development of a symbolic framework for trajectory lifting in covers

Application to a simplified van der Pol oscillator model

Abstract

Chaotic attractors with toroidal topology (van der Pol attractor) have counterparts with symmetry that exhibit unfamiliar phenomena. We investigate double covers of toroidal attractors, discuss changes in their morphology under correlated peeling bifurcations, describe their topological structures and the changes undergone as a symmetry axis crosses the original attractor, and indicate how the symbol name of a trajectory in the original lifts to one in the cover. Covering orbits are described using a powerful synthesis of kneading theory with refinements of the circle map. These methods are applied to a simple version of the van der Pol oscillator.