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 ∈ N, p q, s = (n + 1)p −q >0, n ∈ N, and Fi = (Pi,Qi )t are quasi-homogeneous vector fields of type t = (p, q) and degree i, with Fq−p = (y, 0)t and Qq−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. © 2007 Elsevier Inc. All rights reserved