Stability Of Nontwisted Heteroclinic Loops To 2 Hyperbolic Saddles In Arbitrarily Finite Dimensional Spaces

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 Poincar(?)map is well defined,the criterion is given for the asymptotic stability of the heteroclinic loop confined in its partial neighborhood.The stability results obtained in[1]for 3-dimensional system are extend to(m+n+2)-dimensional space, where m≥0,n≥0.The full paper is divided into three chapters.The first one is devoted to the summarization of the research history and the current situation concerned with the stability of the heteroclinic loop,and to the brief introduction of the contents of the paper.Chapter 2 gives the definitions of the stability for the heteroclinic loop to the hyperbolic saddles in high-dimensional space and hypotheses on the system under consideration.The first section of the third chapter studies the stability of heteroclinic loop to 2 hyperbolic saddles with one dimentional unstable manifold,the second section considers the stability of heteroclinic loop to 2 hyperbolic saddles in general case,the existence and the dimensions of the stable manifold and unstable manifold in its tubular neighborhood are given under different conditions for the leading eigenvalues.Our strategy is as follows.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 orbitΓrespectively,and construct the Poincar(?)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 ofΓ,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.