四次Liénard方程在奇异摄动下极限环的唯一性

The uniqueness of limit cycles for classical Liénard equations of degree four under singular perturbations

Lins-de Melo-Pugh猜想在n=4时已于2012年得到证明,但证明较繁难.本文在奇异摄动框架下对这个结论给出一个简单的证明,借以展示奇异摄动方法在动力系统定性理论研究中的作用,也是本文改进慢发散量积分公式的一个应用.

The Lins-de Melo-Pugh conjecture for n = 4 was proved in 2012, but the proof is complicated. In this article we give a simple proof for this problem under singular perturbations. We want to show the role of singular perturbation method in the study of qualitative theory of dynamical systems, and to show the application of a new formula of the slow divergence integral.