The Study Of Convergence Of N-dimensional Cooperative Systems And Existence Of Limit Cycles In A Cubic System

Monotone dynamical system is the combination of monotone method and dynamical system,it is a new system.At first,Cooperative and compet-itive system of monotone dynamical were discussed by Hirsch[see paper:1-9 and 28].The paper[1-4] exploited the order-preserving properties of coopera-tive system.Paper[5] discussed the existence of equilibria and limit cycles,and global stability and convergence of cooperative or competitive system for three-dimensional.Meanwhile,his paper[7] gived the conclusions about stability and convergence of strongly monotone dynamical systems.Then Smith improved the work of monotone system[see paper:16-27].So,Jiang strenghened the con-diton and conclusion on the basis of previous discussion.In 1989, Hirsch[5] proved the globally stable theorem for three system.Meanwhile,Hirsch pointed out the theorem holds if the conditons became weaker.In 1991,Jiang[8]solved this problem.And Jiang[13] worked out that convergent theorem for three and four-dimensional cooperative system.This paper mainly studied the conver-gence of cooperative system and extended the result of Jiang[13] to the dimen-sion n>4,and the convergent conclusion was as follow:Theorem:For any neZ+ and F:X â†'Rn is cooperative.Suppose the following conditions hold:(a)X(?)Rn be a p-convex and open set;(b)the greatest lower bound and the least upper bound of any compact set in X belongs to X;(c)every equilibrium p ∈ E is Liapunov stable;and(d)every forward semi-orbit has compact closure in X. Then every forward semi-orbit in X converges;i.e.w(x) is a singleton for each x∈X.Therefore,the paper also studied the existence and uniqueness of limit cycle of E1/3 system.Nowadays,many scholars have studied E1/3 system,and got many results.Du Jia[37] exploited the limit cycle and its bifurcation of E1/3 system.Zhou Jiuhong[29] discussed some kinds of E1/3 system.But few people study system of a7≠0.This paper introduced a E1/3 system of a7=0, and analysed the existence and uniqueness of limit cycles.This thesis mainly discussed relative questions for cooperative system and E1/3 system.And it consisted of five parts. The following is the detailed structure of this paper.In chapter one,the historical background of the monotone dynamical sys-tem and E1/3 system were introduced.In the sequel,the study work was summa-rized in the thesis.In chapter two,it gived notation of the basic knowledge and relative the-orems,and illustrated the symbol in this paper.In chapter three,discussed the convergence and divergency of cooperative system.According a kind of special singular points are compared,refining the features of limit sets.And it proved a conjecture of Jiang[13].In chapter four,qualitative properties were analyzed of E1/3 system for α7≠ 0.By introducing Lyapunov formal series,Hopf bifurcation theory and Poincare theorem,the existence and uniqueness of limit cycle were discussed.In chapter five,for relative irreducible system,we had further thoughts and discussion.