A note on the critical periods of potential systems

In this paper, we consider the planar differential system associated with the potential Hamiltonian H(x, y) = (1/2)y2+ V(x) where V(x) = (1/2)x2 + (a/4)x4 + (b/6)x6 with b≠0. This family of differential systems always has a center at the origin and, eventually, three other period annuli. We prove that the corresponding period functions can have at most one critical period altogether. More precisely, we show that this critical period is simple and that it corresponds to the period function associated to the center at the origin. To prove the result we use that the period function verifies a Picard-Fuchs equation. Finally we describe the bifurcation diagram of the period function of the center at the origin. © World Scientific Publishing Company.