Stability of periodic points of diffeomorphism of a plane in the case of a homoclinic orbit

Abstract

We consider a diffeomorphism of a plane into itself with a fixed hyperbolic point at the origin and a non-transversal homoclinic point to it. Periodic points lying in a sufficiently small neighborhood of the homoclinic point are divided into single-pass and multi-pass, depending on the location of the orbit of the periodic point with respect to the orbit of the homoclinic point. From the works of S. Newhouse, L. P. Shil’nikov, B. F. Ivanov and a number of other authors it follows that for a certain method of tangency of the stable and unstable manifold an arbitrarily small neighborhood of the nontransversal homoclinic point can contain an infinite set of stable periodic points, but at least one of the characteristic exponents of these points tends to zero with increasing period. From the previous works of the author it follows that for the other method of tangency of the stable and unstable manifold, an arbitrarily small neighborhood of the nontransversal homoclinic point can contain an infinite set of stable single-pass periodic points. Characteristic exponents of these points are separated from zero. In this paper it is shown that under certain conditions imposed, first of all, on the method of tangency of the stable and unstable manifold, an arbitrarily small neighborhood of the nontransversal homoclinic point can contain a countable set of multi-pass (exactly two-pass) stable periodic points. Characteristic exponents of these points are separated from zero.