一类平面微分系统极限环的存在性和唯一性

Existence and Uniqueness of the Limit Cycles in a Class of Differential System

将对一系列多项式微分系统的定性分析推广到对一类一般的平面微分系统的定性分析,即对一类平面微分系统dxdt=-yf1(x)+δx-lx2n+1dydt=x2n-1f2(x)(其中f1(x),f2(x)∈C1(-∞,+∞))进行定性分析。在适当的条件下,将该系统化为Li啨nard系统进行研究,构造函数λ(x,y)=∫0xg(ξ)dξ+21y2,对λ(x,y)沿着上述微分方程组求导,运用Poincar啨的切性曲线法得到其极限环的不存在性的一系列充分条件。由А.В.Драгилёв存在性定理得到闭轨存在的充分条件,利用O.K.Smith唯一性定理得到极限环存在唯一性的充分条件。

In this paper we generalize the qualitative analysis about a class of plane differential system from the qualitative analysis about a class of polynomial differential system,i.e.the applicable range of this plane differential system will become very wide.Some known results of this class differential system are special events of this plane differential system.Some sufficient conditions for the existence,no-nexistence and uniqueness of the limit cycles for following plane differential system are discussed dxdt=-yf1（x）＋δx-lx2n＋1dydt=x2n-1f2（x） with f1（x）,f2（x）∈C1（-∞,＋∞））On adequacy conditions, we convert this system from plane differential system to Lienard system. We construct a function A （x,y） =∫0xg（ξ）dξ＋21y^2,and differentiate about λ （x,y） with this plane differential system. By using the method of Poincar~, some sufficient conditions for on-existence of limit cycles of such system are ob- tained. By applying A. B. ~par^B theorem about existence, some sufficient conditions for the existence of limit cycles of such system are obtained. Furthermore, by applying O. K. Smith＇s theorem about uniqueness, some sufficient conditions for the uniqueness of limit cycles of such system are obtained.