二次系统极限环的相对位置与个数

<正> 中的P_2(x,y)与Q_2(x,y)为x,y的二次多项式.文1].曾指出,系统(1)最多有三个指标为+1的奇点,且极限环只可能在两个指标为+1的奇点附近同时出现.如果方程(1)的极限环只可能分布在一个奇点外围,我们就说此系统的极限环是集中分布的.本文主要研究具非粗焦点的方程(1)的极限环的集中分布问题,和极限环的最多个数问题.文2]-5]曾证明,当方程(1)有非粗焦点与直线解或有两个非粗焦点或有非粗焦点与具特征根模相等的鞍点时。方程(1)无极限环.本文给出方程(1)具非粗焦点时,极限环集

THE RELATIVE POSITION AND NUMBER OF LIMIT CYCLES OF THE QUADRATIC DIFFERENTIAL SYSTEN

In this paper we study the relative position and number of limit cycles of the quadratic differential system. dx/dt=-y+lx~2+mxy+ny~2,dy/dt=x(1+ax+by)(1)for which O(0,0) is a fine focus. We prove that limit cycles of system (1) Can only be distributed around one of the two focus, provided that system (1) apart from these focus, has a third singuler point.On the other hand system (1) has only two singular points: a fine focus O(0,0) and a rough focus N(0, 1), then (1) may have limit cycles around the two singular points simultaneously. It is important to note that if we choose suitably the Coefficents in system (1), and add a sufficiently small term x to the right hand side of the first equation, there may exist three limit cycles around O(0, 0) and one limit cycle around N(0, 1). Then system (1) has at least four limit cycles.