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

In this paper, we consider C 1 vector fields X in R 3 having a “generalized heteroclinic loop” L which is topologically homeomorphic to the union of a 2–dimensional sphere S 2 and a diameter Γ connecting the north with the south pole. The north pole is an attractor on S 2 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 S 2 . 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 R 3 satisfying this dynamics.