一类具有二虚不变直线的三次系统的极限环与分支

The Bifurcation of Limit Cycle for a Class of Cubic System with Two Imaginary Invariant Line

讨论一类具有二虚平行不变直线的三次系统,求出了奇点O(0,0)的焦点量,证明了δlmn=0时系统在O外围至多有一个极限环.利用分支理论给出了分界线环和半稳定环分支曲线的分支图,进一步说明了系统至多有二个极限环.

This paper considers the bifurcation of limit cycle of a class of cubic system with two imaginary invariant line, and gives the focus valus of each order at O（0, 0）. It is proved that the system with δlmn=0 has at most one limit cycle surrounding O. With the bifurcation theory, the authors give the bifurcation curve of homoclinic cycle and semistable cycle. It means that the system has at most two limit cycles surrounding O.