具有三次曲线解的二次系统的极限环的唯一性  

The Uniqueness of Limit Cycle of a Quadratic System with an Invariant Cubic Curve

作　　者：水树良 

出　　处：《浙江师大学报（自然科学版）》1999年第1期8-11,共4页Journal of Zhejiang Normal University(Natoral Sciences)

摘　　要：本文研究一类以三次曲线ｘｙ２＋２ｙ－１＝０为不变集的二次系统，除去明显不存在极限的情形外，该二次系统可化为ｄｘｄｔ＝（α－１）－（１＋β）ｘ－βｘ２＋αｘｙｄｙｄｔ＝－β２＋（β＋１２）ｙ＋β２ｘｙ－１＋α２ｙ２经一系列变换，将上述方程化为广义Ｌｉéｎａｒｄ方程，证明此方程最多只有一个极限环，从而完整地解决了此类二次系统的极限环的个数问题。In this paper,the author considers a quadratic system with an invariant cubic curve\ xy 2+2y-1=0 .The system can be transformed to: d x d t=(α-1)-(1+β)x-βx 2+αxy d y d t=-β2+(β+12)y+β2xy-1+α2y 2 By some transformations,the system can be changed into a generalized Liénard equation,then prove that the equation has at most one limit cycle.Hence give a satisfactory solution to the question of the number of limit cycle of the discussed system.

关 键 词：二次系统 三次曲线解 极限环 唯一性 

分 类 号：O175[理学—数学]