We study the analytic system of differential equations in the planewhere p, q ∈ N, p q, s = (n + 1)p − q > 0, n ∈ N, and F i = (P i , Q i ) t are quasi-homogeneous vector fields of type t = (p, q) and degree i, with F q−p = (y, 0) t and Q q−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.
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