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

We consider a one-parameter family $(f_{\lambda }{)}_{\lambda ⩾0}$ 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.