具有星形结点的三次系统的极限环  

LIMIT CYCLES OF A PLANAR CUBIC SYSTEM WITH A STAR-SHAPED NODE

作　　者：水树良[1] 蔡燧林[1] 

机构地区：[1]浙江大学

出　　处：《高校应用数学学报（A辑）》1995年第4期361-368,共8页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金

摘　　要：本文研究具有星形结点的三次系统ｘ＝ｘ＋Ｐ２（ｘ，ｙ）＋Ｐ３（ｘ，ｙ），ｙ＝ｙ＋Ｑ２（ｘ，ｙ）＋Ｑ３（ｘ，ｙ）．引入函数ｇ４（θ）（见（１．６））和Ａ（θ）（见（４．４）），得到下述结论：若ｇ４（θ）有零点，则不存在包围原点在其内部的闭轨，特别地，若ｇ４（θ）≡０，则全平面不存在闭轨；若ｇ４（θ）定号，Ａ（θ）常号，则至多存在一个闭轨，若存在，它必包含所有奇点在其内部，且为星形的；若ｇ４（θ）定号而Ａ（θ）变号，则给出了极限环不唯一的例子。In this paper, we consider the limit cycles of a planar cubic system with a star-shaped node x=x+P2(x,y)+P3(x,y), y=y+Q2(x,y)+Q3(x,y).By introducing the functions g4(θ) (see (1. 6)) and A(θ) (see (4. 4)), we obtain the following conclusion: If g4(θ) has a zero point, then the system has no closed orbit surrounding O. Particularly, if g4(θ)≡0, then the system has no closed orbit in whole plane. If g4(θ) does not vanish and A(θ) does not change sign, then the system has at most one limit cycle, and the limit cycle must surround all singular points and must be star-shaped when it exists. We give an example that the system has two limit cycles when g4(θ) does not vanish and A(θ) changes sign.

关 键 词：星形结点 三次系统 极限环 微分方程 

分 类 号：O175.1[理学—数学]