摘要
本文给出关于二次系统极限环唯一性的判别法,它是利用所谓“无切发散量椭园”得到的。设有二次系统,即(Ⅲ)类方程:
In this paper, We consider the quadratic system
(dx)/(dt)=-y+δx+1x^2+mxy+ny^2
(E_2)
(dy)/(dt)=x(1+ax+by)
We propose new criterion for the uniqueness of limit cycle of the system (E_2). Let
N(O)[N(O)] denote the number of limit cycle of (E_2) around the point O(0,0)
[O(0,1/n)]; N(2) denote the number of limit cycle of (E_2). We have
Theorem 1. If following condition are satisfied, then N(0)<1:
(Ⅰ) n(n+b)<0;
(Ⅱ) 1(1b^2-mab+na^2)<0;
(Ⅲ) [m(1+b)-an]~2-41~2n(1+b)<0.
Theorem 2. If conditions of Theorem 1. are satisfied, then N(O)≤1.
Theorem 3. Suppose O(0,0) and O(0,1/n) are focuses then following two conditions are suf-
ficient to insure that N(2)≤2, where N(0)≤1, N(O)≤1:
(Ⅰ) 1(1b^2-mab+na^2)<0;
(Ⅱ) [m(1+b)-an]~2-41~2n(1+b)<0.
出处
《纯粹数学与应用数学》
CSCD
1990年第1期50-54,共5页
Pure and Applied Mathematics
