Generation of symmetric periodic orbits by a heteroclinic loop formed by two singular points and their invariant manifolds of dimensions 1 and 2 in ℝ3

Montserrat Corbera, Jaume Llibre

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Abstract

In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in ℝ3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a "generalized heteroclinic loop" formed by two singular points e+ and e- at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of dimension 1 formed by an orbit going from e- to e+, Γ is contained in ℝ3 and is transversal to Σ. Λ is an invariant manifold of dimension 2 at infinity. In fact, Λ is the two-dimensional sphere at infinity in the Poincaré compactification minus the singular points e+ and e-. The main tool for proving the existence of such periodic orbits is the construction of a Poincaré map along the generalized heteroclinic loop together with the symmetry with respect to Σ. © World Scientific Publishing Company.
Original languageEnglish
Pages (from-to)3295-3302
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume17
DOIs
Publication statusPublished - 1 Jan 2007

Keywords

  • Heteroclinic loops
  • Polynomial vector fields
  • Symmetric periodic orbits

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