Perturbations of quadratic centers

We study the bifurcation of limit cycles in general quadratic perturbations of plane quadratic vector fields having a center at the origin. For any of the cases, we determine the essential perturbation and compute the corresponding bifurcation function. As an application, we find the precise location of the subset of centers in Q(3)(R) surrounded by period annuli of cyclicity at least three. Two specific cases are considered in more detail: the isochronous center S-1 and one of the intersection points (Q(4)(+)) of Q(4) and Q(3)(R). We prove that the period annuli around S-1 and Q(4)(+) have cyclicity two and three respectively. The proof is based on the possibility to derive appropriate Picard-Fuchs equations satisfied by the independent integrals included in the related bifurcation function. (C) Elsevier, Paris.