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

Abstract

In [J. Differential Equations 146 (2) (1998) 320–335], Françoise gives an algorithm for calculating the first nonvanishing Melnikov function M ℓ 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 ω on the family of cycles implies that the form is algebraically relatively exact. We study here a simple example where Françoise's condition is not verified. We generalize Françoise's algorithm to this case and we show that M ℓ belongs to the C[ logt,t,1/t] module above the Abelian integrals. We also establish the linear differential system verified by these Melnikov functions M ℓ( t).