Symmetric periodic orbits near a heteroclinic loop in R³ formed by two singular points, a semistable periodic orbit and their invariant manifolds

In this paper, we consider C1 vector fields X in R3 having a "generalized heteroclinic loop" L which is topologically homeomorphic to the union of a 2-dimensional sphere S2 and a diameter Γ connecting the north with the south pole. The north pole is an attractor on S2 and a repeller on Γ. The equator of the sphere is a periodic orbit unstable in the north hemisphere and stable in the south one. The full space is topologically homeomorphic to the closed ball having as boundary the sphere S2. We also assume that the flow of X is invariant under a topological straight line symmetry on the equator plane of the ball. For each n ∈ N, by means of a convenient Poincaré map, we prove the existence of infinitely many symmetric periodic orbits of X near L that gives n turns around L in a period. We also exhibit a class of polynomial vector fields of degree 4 in R3 satisfying this dynamics. © 2009 Elsevier B.V. All rights reserved.