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

In this paper we will find a continuous of 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 2–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.