高维空间中连接双曲鞍点的异宿环的稳定性

本文考虑任意有限维空间连接两个双曲鞍点的非扭曲异宿环的稳定性问题.在可定义Poincar′e映射的条件下,给出了异宿环在其部分邻域内是渐近稳定的判据,将3维系统鞍点异宿环的稳定性结果推广到了m+n+2维空间中的非扭曲的2-鞍点异宿环,其中m 0,n 0.通过在两个鞍点充分小邻域内,给出系统在适当的线性变换下的第一个规范型,接着采用将局部稳定流形和不稳定流形拉直的变换建立了第二个规范型.然后,在鞍点P1,P2的小邻域内适当选取两个异宿轨道的横截面,并分别分两部分来构造流映射.在鞍点P1,P2的小邻域内,本质上我们利用线性近似系统的流来构造奇异流映射的主部,而在鞍点的邻域外的异宿轨道的小管状邻域内,则用近似于一个非奇异矩阵的微分同胚来获得正则流映射.将四者复合即得到定义于P1小邻域内某横截面上的Poincar′e映射.最后,我们通过技巧性地估计向量的模,给出了在横截面上Poincar′e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,由此得到关于非扭曲2-点异宿环的非常简洁的稳定性判据.

Stability of heteroclinic cycle connecting hyperbolic saddles in higher dimensional space

In this paper, the stability of nontwisted heteroclinic loop to 2 hyperbolic saddles is considered in arbitrarily finite dimensional spaces. Under the condition that the Poincare map is well-defined, the criterion is given for the asymptotic stability of the heteroclinic loop confined in its partial neighborhood. The stability results for 3-dimensional system are extened to m ＋ n ＋ 2 dimensional space, where m≥ 0, n ≥ 0. By taking a suitable linear transformation, we get the first normal form, by a coordinate change to straighten the local stable manifold and the local unstable manifold, we establish the second normal form. Then, in the small neighborhood of the saddle P1, P2, we select two cross sections transversal to the heteroclinic orbit F respectively, and construct the Poincare map by two steps： in the small neighborhood of the saddle, we build the main part of the singular flow map by the linearly approximate system, in the regular tubular neighborhood of F, we use a differential homeomorphism to express the regular flow map. The Poincare map is achieved by composing the singular flow map and the regular flow map. At last, by estimating the modules of some vectors rather skilled, 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 two quite concise stability criteria for the non-resonant heteroclinic cycle to hyperbolic saddles.