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

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.