A generalization of Francoise's algorithm for calculating higher order Melnikov functions

In [J. Differential Equations 146 (2) (1998) 320-335], Francoise gives an algorithm for calculating the first nonvanishing Melmkov function M of a small polynomial perturbation of a Hamiltonian vector field and shows that M-l is given by an Abelian integral. This is done under the condition that vanishing of an Abelian integral of any polynomial form omega on the family of cycles implies that the form is algebraically relatively exact. We study here a simple example where Francoise's condition is not verified. We generalize Francoise's algorithm to this case and we show that Me belongs to the C[log t, t, 1/t] module above the Abelian integrals. We also establish the linear differential system verified by these Melnikov functions M-l (t). (C) 2002 Editions scientifiques et medicales Elsevier SAS. All rights reserved.