Quadratic perturbations of quadratic codimension-four centers

We study the stratum in the set of all quadratic differential systems (x) over dot = P(2)(x, y), (y) over dot = Q(2)(x, y) with a center, known as the codimension-four case Q(4). It has a center and a node and a rational first integral. The limit cycles tinder small quadratic perturbations in the system are determined by the zeros of the first Poincare-Pontryagin-Melnikov integral I. We show that the orbits of the unperturbed system are elliptic curves. and I is a complete elliptic integral. Then using Picard-Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center. (C) 2009 Elsevier Inc. All rights reserved.