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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

Departament de Tecnologies Digitals i de la Informació, Universitat de Vic, 08500 Vic, Barcelona, Spain

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and

JAUME LLIBRE

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

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https://doi.org/10.1142/S0218127407019056Cited by:1 (Source: Crossref)

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.

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