Degenerate lower-dimensional tori in Hamiltonian systems

We study the persistence of lower-dimensional tori in Hamiltonian systems of the form H (x, y, z) (omega, y) + 1/2 (z, M(omega)z) + epsilon P(x, Y, Z, omega), where (x, y, z) is an element of T-n x R-n x R-2m, epsilon is a small parameter, and M(omega) can be singular. We show under a weak Melnikov nonresonant condition and certain singularity removing conditions on the perturbation that the majority of unperturbed n-tori can still survive from the small perturbation. As an application, we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable, properly degenerate Hamiltonian system for which neither the Kolmogorov nor the g-nondegenerate condition is satisfied. (c) 2006 Elsevier Inc. All rights reserved.