Limit cycles for a quadratic perturbation of a quadratic polynomial system

The weak Hilbert 16th problem was solved completely in the quadratic case; that is, the least upper bound of the number of zeros of the Abelian integrals associated with quadratic perturbations of quadratic Hamiltonian systems is known. See [3, 4, 5, 8, 10] and the references therein. The next natural step is to consider the same problem for quadratic integrable but non-Hamiltonian systems. Most mathematicians working in this field believe that perturbations of the quadratic reversible systems, especially with two centers, may give the richest dynamical behavior, see [1] for instance. In this paper we consider quadratic perturbations of a one-parameter family of quadratic reversible non-Hamiltonian systems, with two non-degenerate centers and without other singularities in the finite plane. For such family, we give an upper bound of the limit cycles that emerge from each of the two period annuli and we prove that it is sharp. Even more, we present the exact configuration of limit cycles that can be and the corresponding bifurcation diagrams.