To the question of stability of periodic points of three-dimensional diffeomorphisms

Abstract

We consider diffeomorphism of three-dimensional space with a hyperbolic fixed point at the origin and nontransversal homoclinic point to it. It is follows from the works of Sh. Newhouse, L. P. Shil’nikov, B. F. Ivanov and others that under certain conditions a neighborhood of the homoclinic point contains a countable set of stable periodic points, but at least one of the characteristic exponents at these points tends to zero with increasing period. Earlier, in the author’s work was considered a three-dimensional diffeomorphism and it was assumed that all the eigenvalues of the Jacobi matrix at the origin are real. It was shown that for a certain type of tangency of the stable and unstable manifolds, a neighborhood homoclinic point contains a countable set of stable periodic points with characteristic exponents bounded away from zero. In the present paper we consider the diffeomorphism a three-dimensional space and it is assumed that the Jacobi matrix at the origin has complex eigenvalues. It is shown that in this case the neighborhood nontransversal homoclinic point can contain an infinite number of stable periodic points with characteristic exponents bounded away from zero. Refs 8.