The Existence Of Limit Cycle On Several Differential Systems

The thesis mainly studies several high order differential systems. A group of suffiencit conditions of the existence and uniqueness of limit cycle are obtained. This thesis is composed of four chapters.In Chapter1,the backgroud of related problem and recent develop-ment of the studies in the differential equation are mainly introduced. At the same time,it introduces the task of the paper.In Chapter2, the high order differential systems is discussed, Where a,b,c,σ,l is constant, i,j are odd integers and m>n is integer. The paper makes use of the Poncare segment without contacting, the Theorem of Yushu Xiang to discuss suffiencit conditions of the existence and uniqueness of limit cycle when the paraeters a,b,c,a,l are in varies situations.In Chapter3,the limit cycle of a class of differential systems is discussed,where n> m,2k>2n-2m+1, n, m, k∈N a1, a2, a3are real number and φ(x),φ(x) sastify the following:(H1)φ(x) is even fuction and φ(0)>0; (H2) xφ’(x)>0;(H3) φ(x) is even friction and <p(x)>0. For this system, some sufficient conditions of the existence and nonexis-tence of limit cycle are obtained by using Poincare segment without contacting, the Theorem of Non-intersect and the Theorem of N. Levinson-O.K. Smith.In Chapter4, the limit cycle of a class of differential model is proposed,where φ(x) and φ(x) are even fuctions. In additions, they satisfes the following conditions(1) φ’(0)=φ’’’(0)=φ5(0)…=φ2n+1(0)…=0,(2) φ’(0)=φ’’’0)=φ5(0)…=φ2n+l(0)=0and φ(0)=φ(0)=l. In this Chapter, we have obtained the exstence,uniqueness and nonexistence of limit cycle for the above system by using the First Theorem of Zhangzhifen, the Second Theorem of Zhangzhifen,the Theorem of Hopf Bifucation and so on.