一类平面E_3系统的全局结构

GLOBAL STRUCTURE OF A CLASS OF PLANAR CUBIC SYSTEM(E3)

摘要: 本文研究平面三次系统 (dx)/(dt)=sum from 1≤i+j≤3 a_(i j)X~iy~j (dy)/(dt)=sum from 1≤i+j≤3 b_(i j)X~iy~j E_3 在x~2+y~2=1,y=K_1,y=K_2为(E_3)的代数轨线(可以相交)的假定下,研究了(E_3)全局结构,得出了极限环的存在唯一性及分界线环的存在性,给出了此类系统高阶奇点类型的一种简捷的判别法。

Abstract: A class of planar cubic (E3)with three algebraic orbits x2 + y2 = 1,y = k1, y = k2 is studied. The global structure of (E3) is discussed, the existence and uniqueness of limit cycle and the existence of singular closed orbits are obtained. A easy criteria for determining the type of sigular point of high order is given.