Dissecting a resonance wedge on heteroclinic bifurcations

TL;DR

This paper analyzes complex dynamical behaviors, including chaos and strange attractors, within a resonance wedge in a 3-parameter family of differential equations on a 3-sphere, using return maps and bifurcation theory.

Contribution

It introduces a detailed analysis of heteroclinic bifurcations within a resonance wedge, revealing new dynamical phenomena and extending the understanding of chaos in such systems.

Findings

Identification of a Bogdanov-Takens singularity as organizing center

Existence of infinitely many attracting tori and strange attractors

Connection of resonance wedge structure to classical Arnold tongues

Abstract

This article studies routes to chaos occurring within a resonance wedge for a 3-parametric family of differential equations acting on a 3-sphere. Our starting point is an autonomous vector field whose flow exhibits a weakly attracting heteroclinic network made by two 1-dimensional connections and a 2-dimensional separatrix between two equilibria with different Morse indices. After changing the parameters, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We derive the first return map near the ghost of the attractor and we reduce the analysis of the system to a 2-dimensional map on the cylinder. Complex dynamical features arise from a discrete-time Bogdanov-Takens singularity, which may be seen as the organizing center by which one can obtain infinitely many attracting tori, strange attractors, infinitely many sinks and non-trivial contracting wandering domains. These dynamical phenomena occur within a structure that we call resonance wedge. As an application, we may see the "classical" Arnold tongue as a projection of a resonance wedge. The results are general, extend to other contexts and lead to a fine-tuning of the theory.