Poincaré-Bendixson环域定理的推广  

GENERALIZATION OF THE POINCAR-BENDIXSON'S ANNULAR REGION THEOREM

作　　者：王克[1] 

机构地区：[1]东北师范大学

出　　处：《应用数学学报》1990年第4期401-408,共8页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金

摘　　要：Poincare-Bendixson环域定理是平面动力系统最基本的结论之一,在应用上也极为重要.文献[3]指出,它基本上是由解的存在唯一性和Jordan定理这个简单的几何事实推得的.本文证明,当一个平面微分系统的解不满足唯一性时,Poincare-Bendixson环域定理的结论仍然成立.推广后的环域定理在应用上是方便的.在本文后半部分,我们考虑了Lienard方程的极限环的存在性问题,所得定理推广了著名的定理。The Poincare-Bendixson's annular region theorem is one of the most fundamental resultsin a planar dynamical system, and is also very important in application. But the uniqueness ofsolutions to the system is needed. In this paper we prove that the conclusion of Poincare-Bendi-xson theorem is still valid for a planar differential system without considering the uniqueness ofsolutions. Our theorem is convenient for application. Using our theorem, we consider the exis-tence of limit cycles of Lienard's equation. The well-known Filippov's theorem is generalized.

关 键 词：微分方程组 P-B环域定理 极限环 

分 类 号：O175.12[理学—数学]