LAUSR

Abstract Given m ≥ 1 and a smooth family of planar vector fields ( X e ) e that is a perturbation of a period annulus, we provide a characterization, in terms of Lie brackets, of the property that the first ( m − 1 ) Melnikov functions of ( X e ) e vanish identically. The equivalent condition is the existence of a smooth family of planar vector fields ( U e ) e , called here perturbed normalizers of order m. We also provide an effective procedure for computing U e when the first ( m − 1 ) Melnikov functions of ( X e ) e vanish identically. A formula for the derivative of the m-th order Melnikov function is given.