Stable periodic solutions of periodic systems of differential equations

Abstract

An infinitely differentiable periodic two-dimensional system of differential equations is considered. It is assumed that there is a hyperbolic periodic solution, as well as the presence of a homoclinic solution to the periodic solution. It follows from the works of Sh. Newhouse, L. P. Shil’nikov, B. F. Ivanov and others that under certain conditions a neighborhood of the non-transversal homoclinic solution contains a countable set of stable periodic solutions, but at least one of the characteristic exponents in these solutions tends to zero with increasing period. Earlier, in the author’s work a two-dimensional diffeomorphism was considered and 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. The aim of the present paper is to distinguish a class of two-dimensional periodic systems of differential equations which Poincare transformation is a diffeomorphism that has an infinite set of stable periodic points in the neighborhood of a nontransversal homoclinic point. It is shown that for a certain method of tangency of a stable and unstable manifolds an arbitrary neighborhood of a nontransversal homoclinic solution contains a countable set of stable periodic solutions. Characteristic exponents of these solutions are separated from zero.