Nondegenerate and Nilpotent Centers for a Cubic System of Differential Equations

Abstract

We consider the autonomous system of differential equations of the form $$\begin{aligned} {\dot{x}}=P_1(x,y)+P_2(x,y),\quad {\dot{y}}=Q_1(x,y)+Q_3(x,y), \end{aligned}$$ where \(P_i\) and \(Q_i\) are homogeneous polynomials of degree i. For such systems we provide the necessary and sufficient conditions to have a center at the origin. In fact this family only has nondegenerate and nilpotent centers.