Stable and Completely Unstable Periodic Points of Diffeomorphism of a Plane with a Heteroclinic Contour

Abstract

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