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

摘要

In this paper we consider C1 vector fields X in R3 having a “generalized heteroclinicloop” L which is topologically homeomorphic to the union of a 2–dimensional sphereS2 and a diameter connecting the north with the south pole. The north pole isan attractor on S2 and a repeller on . The equator of the sphere is a periodicorbit unstable in the north hemisphere and stable in the south one. The full spaceis topologically homeomorphic to the closed ball having as boundary the sphereS2. We also assume that the flow of X is invariant under a topological straight linesymmetry on the equator plane of the ball. For each n ∈ N, by means of a convenientPoincar´e map, we prove the existence of infinitely many symmetric periodic orbitsof X near L that gives n turns around L in a period. We also exhibit a class ofpolynomial vector fields of degree 4 in R3 satisfying this dynamics.