The Successive Derivatives of the Period Function of a Plane Vector Field

Previously, we provided an expression which generalized the classical Melnikov function to any order, for the first nonzero derivative of a return mapping. Our method relied on the decomposition of a 1-form associated to the relative cohomology of the perturbed Hamiltonian. With the same techniques, we give a formula for the first nonzero derivative of a period function. We focus on the particular example ofH=(1/2)(x2+y2) and then we define a class of Hamiltonians for which the same computation remains valid. Finally, we investigate relations with Birkhoff normal form