一类具有二虚不变直线的三次系统的极限环  

Limit cycle for a class of cubic system with two imaginary invariant lines

作　　者：郑燕花 谢向东[2] 

机构地区：[1]厦门康桥中学,福建厦门361012 [2]宁德师范高等专科学校数学系,福建宁德352100

出　　处：《高校应用数学学报（A辑）》2009年第1期47-52,共6页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：福建省自然科学基金(Z0511052)

摘　　要：研究一类具有二虚不变直线的三次系统:x′=y(1+x^2),y′=-x+δy+nx^2+ mxy+ly^2+bxy^2,分析奇点的性态并求出奇点O的焦点量w_0=δ,w_1=m(n+l),w_2= -mn(b-1).证明了w_0=w_1=w_2=0时O为中心,并证明了w_0=0,w_1w_2≥0时系统无极限环;w_0=0,w_1w_2<0时系统至多有一个极限环.A class of cubic system with two imaginary invariant line x′= y（1 ＋ x^2）,y′ = -x ＋δy ＋ nx^2 ＋ mxy ＋ ly^2 ＋ bxy^2 is studied. The character of the critical point is studied and the successive focal values w0 = δ, w1 = m（n ＋ l）, w2 = -mn（b - 1）are obtained. It is proved that if w0 = w1 = w2 = 0, then O is a center. It is also proved that the system has no limit cycles whenwo = 0, w1w2 ≥0, and has at most one limit cycle when w0 = 0, w1w2 〈 0.

关 键 词：不变直线 三次系统 极限环 

分 类 号：O175.12[理学—数学]