Stability of heteroclinic cycle connecting hyperbolic saddles in higher dimensional space

Abstract

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′e 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 Γ respectively, and construct the Poincar′e 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 Γ, we use a differential homeomorphism to express the regular flow map. The Poincar′e 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.