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.

中文翻译：

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具有等斜轮廓的微分方程周期系统周期解的稳定性

摘要

考虑具有两个双曲周期解的微分方程的二维周期系统。假定异斜解位于固定点的稳定流形和不稳定流形的相交处。更准确地说，假设存在一个非等速轮廓。我研究了稳定歧管和不稳定歧管在至少一个异斜解的点处非横向相交的情况。有多种方法可以将非稳态歧管与稳定歧管非横向相交。早些时候，在LP Shilnikov，SV Gonchenko，BF Ivanov等人的著作中，有人提出，在稳定流形和不稳定流形的非横向交点处，切线不超过有限阶。从这些作者的工作可以得出，在异斜线轮廓附近存在稳定的周期解的系统。在本文中，在以下假设下研究了等斜线轮廓：在等斜线段上的切线不是有限阶切线，切线在稳定和不稳定歧管的非横向相交点处。结果表明，一组可数的周期解位于这种等斜线轮廓的附近，该等斜线轮廓的特征指数与零分离。在非斜解的点处的稳定和不稳定歧管的非横向交点处，相切不是有限阶的相切。结果表明，一组可数的周期解位于这种等斜线轮廓的附近，该等斜线轮廓的特征指数与零分离。在非斜解的点处的稳定和不稳定歧管的非横向交点处，相切不是有限阶的相切。结果表明，一组可数的周期解位于这种等斜线轮廓的附近，该等斜线轮廓的特征指数与零分离。