Symmetric periodic orbits near a heteroclinic loop in R3R3 formed by two singular points, a semistable periodic orbit and their invariant manifolds

In this paper, we consider C1C1 vector fields XX in R3R3 having a “generalized heteroclinic loop” LL which is topologically homeomorphic to the union of a 2–dimensional sphere S2S2 and a diameter ΓΓ connecting the north with the south pole. The north pole is an attractor on S2S2 and a repeller on ΓΓ. 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 S2S2. We also assume that the flow of XX is invariant under a topological straight line symmetry on the equator plane of the ball. For each n∈Nn∈N, by means of a convenient Poincaré map, we prove the existence of infinitely many symmetric periodic orbits of XX near LL that gives nn turns around LL in a period. We also exhibit a class of polynomial vector fields of degree 4 in R3R3 satisfying this dynamics.