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

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.