The role of the saddle-foci on the structure of a Bykov attracting set

TL;DR

This paper investigates how saddle-focus dynamics influence the structure of a heteroclinic network on the 3-sphere, revealing the emergence of complex invariant sets and their stability within the global attractor.

Contribution

It demonstrates that for small positive parameters, the invariant manifolds and horseshoes are contained in the global attractor and belong to the heteroclinic class, extending understanding of saddle-focus heteroclinic networks.

Findings

Invariant manifolds are contained in the global attractor.

Horseshoes belong to the heteroclinic class of the equilibria.

The set of chain-accessible points is chain-stable and contains the closure of invariant manifolds.

Abstract

We consider a one-parameter family $(f_\lambda)_{\lambda \, \geqslant \, 0}$ of symmetric vector fields on the three-dimensional sphere $\mathbb{S}^3\subset\mathbb{R}^4$ whose flows exhibit a heteroclinic network between two saddle-foci inside a global attracting set. More precisely, when $\lambda = 0$, there is an attracting heteroclinic cycle between the two equilibria which is made of two $1$-dimensional connections together with a $2$-dimensional sphere which is both the stable manifold of one saddle-focus and the unstable manifold of the other. After slightly increasing the parameter while keeping the $1$-dimensional connections unaltered, the two-dimensional invariant manifolds of the equilibria become transversal, and thereby create homoclinic and heteroclinic tangles. It is known that these newborn structures are the source of a countable union of topological horseshoes, which prompt the coexistence of infinitely many sinks and saddle-type invariant sets for many values of $\lambda$. We show that, for every small enough positive parameter $\lambda$, the stable and unstable manifolds of the equilibria and those infinitely many horseshoes are contained in the global attracting set of $f_\lambda$. Moreover, we prove that the horseshoes belong to the heteroclinic class of the equilibria. In addition, we verify that the set of chain-accessible points from either of the saddle-foci is chain-stable and contains the closure of the invariant manifolds of the two equilibria.