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

• Bessa Mário ^[1] ; Carvalho, Maria ^[2] ; Rodrigues Alexandre A P ^[2]
  1. [1] Universidade da Beira Interior

     Universidade da Beira Interior

     Covilhã (Conceição), Portugal

     
  2. [2] Universidade Do Porto

     Universidade Do Porto

     Santo Ildefonso, Portugal

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
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• Resumen

  • 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.