The Role of the Saddle-Foci on the Structure of a Bykov Attracting Set

We consider a one-parameter family $$(f_\lambda )_{\lambda \, \geqslant \, 0}$$ of symmetric vector fields on the three-dimensional sphere 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 saddle-foci and those infinitely many horseshoes are contained in the global attracting set of $$f_\lambda $$; moreover, the horseshoes belong to the heteroclinic class of the equilibria. In addition, we show 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.