Stability Of Homoclinic Loop To Saddle-Focus

In recent years, an increasingly more attention has been paid to the bifurcation problems and the chaos phenomena accompanied with the homoclinic loop. In solving this kind of problems, the stability of the homoclinic loop itself plays an important role. The research fruits concerned with the stability of the plane homoclinic loop have been very rich. There are also some researches on the stability of the homoclinic loop to the saddle point in the space. However, due to the difficulties in the study, there has been no research on the stability of the homoclinic loop to saddle-focus in the space with dimension larger than 3.In this paper , the stability is investigated deeply for the homoclinic loop to a hyperbolic saddle-focus in arbitrarily finite dimensional spaces. The paper consists of two parts. In the first part, we outline the history and current development of the stability study of the homoclinic loop. Then a brief summarization of this paper is given. In the second part,firstly we study the stability for the homoclinic orbit to a hyperbolic saddle-focus with 1-dimensional unstable manifold in higher dimensional systems. Secondly, the dimensions of the stable manifold and the unstable manifold confined in some partial neighborhood of the homoclinic orbit to a general hyperbolic saddle-focus are investigated for arbitrarily finite-dimensional systems.At first, we study the stability of the homoclinic orbit to a hyperbolic saddle-focus with 1-dimensional unstable manifold in higher dimensional systems. Consider the following (m+3)-dimensional systemAssume system (2.1) has a homoclinic orbit to a hyperbolic saddle-focus. In order to study the stability for the homoclinic orbit to a hyperbolic saddle-focus better, we improve and generalize the stability definition first given by [11] to make it be adaptable to any space homoclinic loop confined in some partial neighborhood.The following two hypotheses are fulfilled by system (2.1) .(H1) O is a hyperbolic saddle-focus of system (2.1), more precisely,F(O)= 0,α±βi,λ^* andλ_i=1,2,…,m are the eigenvalues of A = D_xF(O)satisfyingα<0,β>0,λ^*>0å’ŒRe{λ₁,λ₂,…,λ_m}<α<0.(H2) System (2.1) has a homoclinic orbitΓto the equilibrium O. Let E^s and E^u be the subspace, respectively, spanned by the eigenvectors correspondingto the eigenvaluesα±βi andλ^*,then the positive half-orbit ofΓis tangent to E^s at point O,and the negative half-orbit ofΓis tangent to E^u at point O. Under the above assumptions, we give a criterion for the asymptotical stability of homoclinic cycles to a saddle-focus. It takes us six steps to prove this theorem. We first show five auxiliary lemmas, then give a direct proof of the theorem.Notice that in the section 3.2cii of chapter 3 of book [13], the same stability conclusion is given to our paper in the particular case m=0. But, first, its proving method has some problems. Second, this kind of stability conclusion is incorrect whatever under general stability definition or the stability definition in some partial neighborhood first given by [11], but only true after improving and generalizing the stability definition.Then second, we discuss the stability of the homoclinic loop to a hyperbolic saddle-focus in arbitrarily finite dimensional spaces.Considering (m+n+3) -dimension systemStill assume that system (2.10) has a homoclinic orbit to a hyperbolic saddle-focus. By improving and generalizing the stability definition adaptable to the space homoclinic loop confined in some partial neighborhood, we give further the definition of infinitive-Cantor set and the definition of the stable set (manifold) and unstable set (manifold) in its tubular neighborhood.For system (2.10),we make the following assumptions:(A1) F(0)=0,α±βi,λ^* andλ_ui,i=1,2,…,m andλ_vj,j=1,2,…,n are theeigenvalues of A = D_xF(0) satisfyingα<0,β>0,λ^*>0and Re{λ_u1,λ_u2,…,λ_um}<α<0,Re{λ_v1,λ_v2,…,λ_vn}>λ^*>0.(A2) System (2.1) has a homoclinic orbit T to fixed point O.Let E^s and E^u be the subspace, respectively, spanned by the eigenvectors correspondingto the eigenvaluesα±βi andλ^*,then the positive half-orbit ofΓis tangent toE^s at point O,and the negative half-orbit ofΓis tangent to E^u at point O.Under the above assumptions and definitions, by establishing the first recurrent map, we study the existence and the dimensions of the stable set (manifold) and unstable set (manifold) in its tubular neighborhood and get the main result of this paper.