具有二阶细焦点的二次系统极限环的唯一性

ON THE UNIQUENESS OF THE LIMIT CYCLE OF THE QUADRATIC SYSTEM WITH A WEAK FOCUS OF ORDER 2

人们猜想,平面二次系统二阶细焦点外围至多存在一个极限环,但迄今未能证实.文[1,2]在某些参数取特定值之下证明了这一猜想.最近文[5]在具有零特征根奇点之下也证明了这一猜想.本文则在较一般的情况下证明了这一猜想,并使文[5]的结果成为本文的特例.此外,本文还给出了若干有环无环的条件.

This paper,under fairly general conditions,it is proved that a quadratic system with afocus of order 2 has at most one limit cycle around the weak focus.As a consequence,all theresults of [5] can be deduced from this paper.