一类具细焦点的三次系统极限环的唯一性

The Uniqueness of Limit Cycle for a Class of Cubic System with Weak Focus

继续相关文献的工作,给出与二次系统Ⅰ相伴的一类三次系统在奇点N(0,1/n)的焦点量公式,证明了系统在细焦点N外围至多有一个极限环,同时证明了当N或O为细焦点时,系统在另一个焦点外围无极限环,结合相关文献的结论,说明了具有细焦点的该系统在全平面至多有一个极限环.

This paper we continue the study of paper [1] and give the focus valus of each order at N（0, 1/n） for a class of accompany system of quadratic differential system I. We have proved that the system has at most one limit cycle surrounding weak focus N,and have proved that if the system （2） has one weak focus, then it has at most one limit cycles surrounding other focus. It means that the system （2） with weak focus has at most one limit cycles in the plane.