Unfolding a Bykov attractor: from an attracting torus to strange attractors

TL;DR

This paper investigates how complex chaotic attractors emerge from a heteroclinic network in a three-dimensional system, revealing a transition from tori to strange attractors through parameter variation.

Contribution

It provides a detailed mechanism for the emergence of strange attractors from a Bykov heteroclinic network in a 3D differential system, extending the understanding of bifurcations leading to chaos.

Findings

Existence of attracting quasi-periodic tori.

Presence of Hénon-like strange attractors.

Application to unfoldings of Hopf-zero singularities.

Abstract

In this paper we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two hyperbolic saddles-foci with different Morse indices. After slightly increasing both parameters, while keeping the one-dimensional connections unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the equilibria do not intersect. Under some conditions on the parameters and on the eigenvalues of the linearisation of the vector field at the saddle-foci, we prove the existence of many complicated dynamical objects, ranging from an attracting quasi-periodic torus to H\'enon-like strange attractors, as a consequence of the Torus-Breakdown Theory. The mechanism for the creation of horseshoes and strange attractors is also discussed. Theoretical results are applied to show the occurrence of strange attractors in some analytic unfoldings of a Hopf-zero singularity.