The center problem for a family of systems of differential equations having a nilpotent singular point

We study the analytic system of differential equations in the plane(x˙,y˙)t=∑i=0∞Fq−p+2is, where p,q∈Np,q∈N, p⩽qp⩽q, s=(n+1)p−q>0s=(n+1)p−q>0, n∈Nn∈N, and Fi=t(Pi,Qi)Fi=(Pi,Qi)t are quasi-homogeneous vector fields of type t=(p,q)t=(p,q) and degree i  , with Fq−p=t(y,0)Fq−p=(y,0)t and Qq−p+2s(1,0)<0Qq−p+2s(1,0)<0. The origin of this system is a nilpotent and monodromic isolated singular point. We prove for this system the existence of a Lyapunov function and we solve theoretically the center problem for such system. Finally, as an application of the theoretical procedure, we characterize the centers of several subfamilies.