连接两个具有一维不稳定流形的双曲鞍点异宿环的稳定性  

The Stability of Heteroclinic Loop Connecting Two Hyperbolic Saddles with One Dimensional Unstable Manifold

作　　者：路秋英 邓桂丰 刘潇[2] 朱德明[2] Qiu Ying LU;Gui Feng DEN;Xiao LIU;De Ming ZHU(Shanghai Lixin University of Accounting and Finance,Shanghai 201209,P.R.China;Department of Mathematics,East China Normal University,Shanghai 200241,P.R.China)

机构地区：[1]上海立信会计金融学院统计与数学学院,上海201209 [2]华东师范大学数学系,上海200241

出　　处：《数学学报（中文版）》2018年第5期761-770,共10页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11101370,11211130093)

摘　　要：本文研究任意有限维空间中连接两个具有一维不稳定流形的双曲鞍点异宿环的稳定性.借助适当的线性变换和坐标变换,将局部稳定流形和不稳定流形拉直,利用奇异流映射和正则流映射构造了Poincaré映射.通过技巧性地估计向量的模,给出了在横截面上Poincaré映射的初始点与首次回归点离异宿轨道与横截面交点的距离之比,得到了高维空间中连接两个带有一维不稳定流形的异宿环的非常简洁的稳定性判据.In this paper, the stability of heteroclinic loop connecting two hyperbolic saddles with one dimensional unstable manifold is considered in arbitrarily finite dimensional spaces. By taking a suitable linear transformation and coordinate change to straighten the local stable manifold and the local unstable manifold, we construct the Poincaré map by composing the singular flow map and the regular flow map. By estimating the modules of some vectors rather skillfully, we obtain the ratio of the distance between the first recurrent point and the heteroclinic point to the distance between the initial point and the heteroclinic point. As a direct consequence, we derive the concise stability criterion of the heteroclinic loop connecting two hyperbolic saddles with one-dimentional unstable manifold for the higher dimensional system.

关 键 词：高维系统 异宿环 稳定性 POINCARÉ映射 

分 类 号：O175.1[理学—数学]