一类阿贝尔方程极限环的几个判定准则  

Several Criterions to Determine the Limit Cycles of a Class of Abel Equations

作　　者：张剑峰[1] 陈晓星[1] 

机构地区：[1]福州大学数学系

出　　处：《福州大学学报（自然科学版）》1999年第1期9-11,共3页Journal of Fuzhou University(Natural Science Edition)

基　　金：国家自然科学基金

摘　　要：当ｎ次微分系统ｘ′＝λｘ－ｙ＋Ｐｎ（ｘ，ｙ），ｙ′＝ｘ＋λｙ＋Ｑｎ（ｘ，ｙ）（ｎ≥２）化为Ａｂｅｌ方程ｄｚ／ｄθ＝Ａ（θ）ｚ３＋Ｂ（θ）ｚ２＋Ｃ（θ）ｚ后，利用λＡ（θ）的符号给出了判定Ａｂｅｌ方程极限环的几个准则：（１）当λＡ（θ）≥０且ｎ为偶数时方程无极限环；（２）若λＡ（θ）≤０时，则方程存在唯一极限环；（３）若λＡ（θ）≥０且ｎ为奇数，则方程最多只有两个极限环．After transforming the n -degree differential systems x′=λx-y+P n(x,y), y′=x+λy+Q n(x,y)(n≥2) into Abel equations d z/ d θ=A(θ)z 3+B(θ)z 2+C(θ)z , we obtain several criterions to determine the limit cycles of the Abel equations according to the sign of λA(θ) :(1)when λA(θ) ≥0 and n be even, the equations have no limit cycles. (2)when λA(θ) ≤0,the equations have unique limit cycles. (3)if λA(θ) ≥0 and n be odd, the equations have at most two limit cycles.

关 键 词：微分系统 阿贝尔方程 极限环 判定准则 

分 类 号：O175.12[理学—数学]