Bifurcation of limit cycles from two families of centers

We study the number of limit cycles that bifurcate from the periodic orbits of a center in two families of planar polynomial systems. One of these families has a global center. The other family is obtained by adding a straight line of critical points to the first one. The common point between both unperturbed families is that they can be integrated by using the Lyapunov polar coordinates. The study of the number of limit cycles bifurcating from the centers is done by considering the zeros of the associated Poincare-Melnikov integrals. As a consequence of our study we provide quadratic lower bounds for the number of limit cycles surrounding a unique critical point in terms of the degree of the system.