一类三次系统两个中心共存的条件

对于一类平面三次系统dx/dt=y a1x^2 (a2 2b1)xy (a3-a1)y^2 xf(x,y),dy/dt=-x b1x^2 (b2-2a1)xy-b1y^2 yf(x,y),其中f(x,y)=a1x^2 a5xy (a6-a1)y^2.N.G.Lloyd,C.J.Christopher等研究了系统(1)的原点是中心的充要条件．除原点O(0，0)之外，如果系统(1)还存在另一奇点，它是中心或焦点型的(即在奇点处一次近似系统为中心)，本文讨论系统(1)的两个中心共存的条件．

Conditions for certain cubic systems to have two centers

In this paper, the quadratic-like cubic systems dxdt=y+a_1x~2+(a_2+2b_1)xy+(a_3-a_1)y~2+xf(x,y),dydt=-x+b_1x~2+(b_2-2a_1)xy-b_1y~2+yf(x,y), (1)in which f(x,y)=a_4x~2+a_5xy+(a_6-a_4)y~2 is considered. N.G.Lloyd and C.J.Christopher et al.studied the sufficient conditions for that, the origin of the system(1) is a center. Our interest is the coexistence of two centers for the system(1). The conditions for the system(1) to have two centers is also briefly given.