Torus-breakdown near a Bykov attractor: a case study

TL;DR

This paper explores complex dynamics and bifurcations in a symmetric vector field on -sphere, revealing how invariant tori break down and lead to strange attractors through numerical experiments and theoretical analysis.

Contribution

It provides the first explicit numerical analysis of torus breakdown near a Bykov attractor in a symmetric vector field, linking bifurcations to strange attractors.

Findings

Detection of strange attractors via Torus-Breakdown theory

Identification of invariant manifold changes leading to torus destruction

Insights into bifurcation mechanisms near a Bykov attractor

Abstract

There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of vector fields unfolding an attracting heteroclinic network, linking two saddle-foci with $ (\mathbb{SO}(2) \oplus \mathbb{Z}_2)$-symmetry. The vector field is the restriction to $\mathbb{S}^3$ of a polynomial vector field in $\mathbb{R}^4$. We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a phenomenon called Torus-Breakdown theory. We explain how an attracting torus gets destroyed by following the changes in the invariant manifolds of the saddle-foci. Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is still out of reach, using a combination of theoretical tools and computer simulations, we have uncovered some complex patterns for the symmetric family under analysis. This also suggests a route to obtain rotational horseshoes; additionally, we give an attempt to elucidate some of the bifurcations involved in an Arnold wedge.