二次系统(Ⅲ)_(n=0)一阶细焦点外围极限环的惟一性

ON THE UNIQUENESS OF LIMIT CYCLES OF QUADRATIC SYSTEM OF TYPE (Ⅲ)_(n=0)

本文证明二次系统 ( ) n=0 方程当其细焦点的一阶细焦点量 (w1 )和三阶细焦点量 (w3)的符号异号时 ,该细焦点外围至多有一个极限环 ;当 w1 与 w3符号相同时 ,该细焦点外围可以出现二个极限环 ,并举出了例子 .

In this paper,it is proved that if the product of the first focus value w 1 and third focus value w 3 of the weak focus for quadratic system of type(Ⅲ) n=0 is negative,then the system has at most one limit cycle surrounding the weak focus;if w 1w 3>0,then some of the systems have two limit cycles around the weak foucs. An example is given.