Stability of periodic solutions of periodic systems of differential equations with a heteroclinic contour

Abstract

A two-dimensional periodic system of differential equations with two hyperbolic periodic solutions is considered. It is assumed that heteroclinic solutions lie at the intersection of stable and unstable manifolds of fixed points; more precisely, the existence of a heteroclinic contour is assumed. I study the case in which stable and unstable manifolds intersect nontransversally at the points of at least one heteroclinic solution. There are various ways of nontransversally intersecting a stable manifold with an unstable manifold at the points of a heteroclinic solution. Earlier, in the works of L.P. Shilnikov, S.V. Gonchenko, B.F. Ivanov, et al., it was suggested that, at the points of nontransversal intersection of a stable and an unstable manifold, there is a tangency of no more than finite order. It follows from the works of these authors that there exist systems in which there are stable periodic solutions in the neighborhood of the heteroclinic contour. In this paper, heteroclinic contours are studied under the assumption that, at the points of nontransversal intersection of a stable and an unstable manifold at the points of the heteroclinic solution, the tangency is not a tangency of finite order. It is shown that a countable set of periodic solutions is situated in the neighborhood of such a heteroclinic contour the characteristic exponents of which are separated from zero.