具有两个相切圆解的三次系统的极限环

Limit Cycles of a Cubic System with Two Tangent Circle Solutions

黄启宇给出了三次系统具有两个相分离的圆解为极限环的充要条件,并证明当两圆解成为极限环时,不存在任何其它极限环。本文进一步证明了具有两个相分离解的三次系统无论两圆解是否为极限环,都不会有其它极限环存在,而具有两个相切圓解的三次系统是可以存在其它极限环的。

Paper [1] gave out the necessary and sufficient codition for the existence of two separated circle solutions as its limit cycles in a cubic system, and then proved that there is no other limit cycles when the two circle solutions become limit cycles. In fact, in a cubic system there exists no other limit cycle, no matter whether the two circle solutions are limit cycles or not. In this paper, we prove that other limit cycle may exist for the cubic system with two tangent circle solutions.