Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit

Abstract

A diffeomorphism of the plane into itself with a fixed hyperbolic point is considered; the presence of a nontransverse homoclinic point is assumed. Stable and unstable manifolds touch each other at a homoclinic point; there are various ways of touching a stable and unstable manifold. In the works of Sh. Newhouse, L. P. Shilnikov and other authors, studied diffeomorphisms of the plane with a nontranverse homoclinic point, under the assumption that this point is a tangency point of finite order. It follows from the works of these authors that an infinite set of stable periodic points can lie in a neighborhood of a homoclinic point; the presence of such a set depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point at which the tangency of a stable and unstable manifold is a tangency of finite order. Allocate a countable number of types of periodic points lying in the vicinity of a homoclinic point; points belonging to the same type are called n-pass (multi-pass), where n is a natural number. In the present paper, it is shown that if the tangency is not a tangency of finite order, the neighborhood of a nontransverse homolinic point can contain an infinite set of stable single-pass, double-pass, or three-pass periodic points with characteristic exponents separated from zero.

Keywords:

diffeomorphism, nontransverse homoclinic point, stability, characteristic exponents