On the number of limit cycles in quadratic perturbations of quadratic codimension-four centres

This paper is concerned with the bifurcation of limit cycles in general quadratic perturbations of quadratic codimension-four centres Q(4). Gavrilov and Iliev set an upper bound of eight for the number of limit cycles produced from the period annuli around the centre. Based on Gavrilov-Iliev's proof, we prove in this paper that the perturbed system has at most five limit cycles which emerge from the period annuli around the centre. We also show that there exists a perturbed system with three limit cycles produced by the period annuli of Q(4).