On the period function of centers in planar polynomial Hamiltonian systems of degree four

In this paper we study non-degenerate centers of planar polynomial Hamiltonian systems. We prove that if the differential system has degree four then the period function of the center tends to infinity as we approach to the boundary of its period annulus. The proof takes advantage of the geometric properties of the period annulus in the Poincaŕe disc and it requires the study of the so called cubic-like Hamiltonian systems, namely the differential systems associated to a Hamiltonian function of the form H(x, y) = A(x) + B(x)y + C(x)y2 + D(x)y3. Concerning the centers of this family of differential systems, we obtain an analytic expression of its period function. From our point of view this expression constitutes the first step in order to find the isochronicity conditions in the family.