Stable and completely unstable periodic points of diffeomorphism of a plane with a heteroclinic contour

Abstract

We study the diffeomorphism of a plane into itself with three fixed hyperbolic points. It is assumed that at the intersections of the unstable manifold of the first point and the stable manifold of the second point, the unstable manifold of the second point and the stable manifold of the third point, the unstable manifold of the third first point and the stable manifold of the first point are heteroclinic points. The orbits of fixed and heteroclinic points form a heteroclinic contour. The case is studied when stable and unstable manifolds intersect non-transversally at heteroclinic points. Among the points of non-transversal intersection of a stable manifold with an unstable manifold, first of all, points of tangency of finite order are distinguished; in this paper, such points are not considered. In the works of L. P. Shilnikov, S. V. Gonchenko and other authors studied diffeomorphism with heteroclinic contour, it was assumed that the points of non-transversal intersection of a stable and unstable manifolds are points of tangency of finite order. It follows from the works of these authors that there exist diffeomorphisms for which there are stable and completely unstable periodic points in the neighborhood of the heteroclinic contour. In this paper, it is assumed that the points of a non-transversal intersection of a stable and unstable manifolds are not points of tangency of finite order. It is shown that in the neighborhood of such a heteroclinic contour two countable sets of periodic points can lie. One of these sets consists of stable periodic points, whose characteristic exponents are separated from zero, the second — from completely unstable periodic points, whose characteristic exponents are also separated from zero.