Perturbed normalizers and Melnikov functions

Given m≥1 and a smooth family of planar vector fields (Xε)ε 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 (Xε)ε vanish identically. The equivalent condition is the existence of a smooth family of planar vector fields (Uε)ε, called here perturbed normalizers of order m. We also provide an effective procedure for computing Uε when the first (m−1) Melnikov functions of (Xε)ε vanish identically. A formula for the derivative of the m-th order Melnikov function is given.