Existence of at most 1, 2, or 3 zeros of a Melnikov function and limit cycles

We investigate the existence of at most one, two, or three limit cycles bif urcated from a periodic annulus of a Hamiltonian system under a class of pe rturbations and obtain some sufficient conditions which ensure that the cor responding Melnikov function has at most one, two, or three zeros in an ope n interval. We also give applications to some systems which appear in codim ension two bifurcations and to some Lienard systems. (C) 2001 Academic Pres s