研究了一类三次多项式微分系统$\dot{x}=-y+\delta x+l{x}^2+mxy+bx{y}^2+a{x}^3,\dot{y}=x$ 的广义相伴系统$\dot{x}=-y+\delta x+l{x}^2+mxy+bx{y}^2+a{x}^3,\dot{y}=x\phi (y)$, 对原点$O$ 进行了中心-焦点判定. 利用旋转向量场的理论得出了系统不存在极限环的充分条件, 利用Hopf 分支问题的Lyapunov 第二方法得到了该系统极限环 存在性的若干充分条件, 最后利用Coppel的唯一性定理得到了极限环唯一性的充分条件.

Studied a class of generalized accompanying system x˙ = −y +x+lx2 + mxy + bxy2 + ax3, y˙ = x'(y) of the cubic system x˙ = −y + x + lx2 + mxy + bxy2 + ax3, y˙ = x. The origin is the center or focus is judged. Using the theory of a rotating vector field, the sufficient conditions for no-existence of limit cycles are obtained. By using the second method of Lyapunov for Hopf bifurcation problem, some sufficient conditions for the existence of limit cycle of the system are obtained. By using the uniqueness theorem of Coppel for limit cycles, some sufficient conditions for the uniqueness of limit cycle are obtained.