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

• Autor(a) principal: Bessa, Mário
• Data de Publicação: 2020
• Outros Autores: Carvalho, Maria, Rodrigues, Alexandre A. P.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.2/13896
• topic: Heteroclinic cycle; Bykov network; Chain-accessible; Chain-recurrent; Symmetry

We consider a one-parameter family ( fλ)λ 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 λ = 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 λ. We show that, for every small enough positive parameter λ, the stable and unstable manifolds of the saddle-foci and those infinitely many horseshoes are contained in the global attracting set of fλ; 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.