Abstract

The diffeomorphism of a plane into itself with three hyperbolic points is studied this paper. It is assumed that the heteroclinic points lie at the intersections of the unstable manifold of the first point and the stable manifold of the second point, of the unstable manifold of the second point and the stable manifold of the third point, of the unstable manifold of the third point and the stable manifold of the first point. The orbits of fixed and heteroclinic points form a heteroclinic contour. The case when stable and unstable manifolds intersect non-transversally at heteroclinic points is investigated. The points of tangency of finite order are firstly distinguished among the points of non-transversal intersection of a stable manifold with an unstable manifold; in this paper, such points are not considered. Diffeomorphism with a heteroclinic contour was studied in the works of L.P. Shilnikov, S.V. Gonchenko, and other authors, and it was assumed that the points of non-transversal intersection of stable and unstable manifolds are points of tangency of finite order. It follows from the works of these authors that a diffeomorphism exists for which there are stable and completely unstable periodic points in the neighborhood of the heteroclinic contour. It is assumed in the current paper that the points of non-transversal intersection of stable and unstable manifolds are not the points of tangency of finite order. It is demonstrated that two countable sets of periodic points may lie in the neighborhood of such a heteroclinic contour. One of these sets consists of stable periodic points whose characteristic exponents are separated from zero, and another set consists of completely unstable periodic points whose characteristic exponents are also separated from zero.

中文翻译：

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具有斜面轮廓的平面的微差同构的稳定且完全不稳定的周期点

摘要

本文研究了一个具有三个双曲点的平面的微分。假设非斜点位于第一点的不稳定歧管和第二点的稳定歧管，第二点的不稳定歧管和第三点的稳定歧管，第三点和第一点的稳定流形。固定点和非斜点的轨道形成一个非斜轮廓。研究了稳定歧管和不稳定歧管在异斜点非横向相交的情况。首先在有限流形与不稳定流形的非横向相交点之间区分有限阶切点。在本文中，未考虑这些要点。在LP Shilnikov，SV Gonchenko等人的著作中研究了具有异斜线轮廓的微分同构，并假设稳定和不稳定流形的非横向交点为有限阶切点。从这些作者的著作中可以得出，存在一个微分同构，对于该同构，在异斜等高线附近存在稳定和完全不稳定的周期点。本文假设，稳定流形和不稳定流形的非横向相交点不是有限阶的相切点。证明了两个可数的周期点集可能位于这种异斜线轮廓的附近。其中一组包含稳定的周期点，其特征指数与零分开，