Degenerate lower-dimensional tori in Hamiltonian systems

Abstract

We study the persistence of lower-dimensional tori in Hamiltonian systems of the form H ( x , y , z ) = 〈 ω , y 〉 + 1 2 〈 z , M ( ω ) z 〉 + ε P ( x , y , z , ω ) , where ( x , y , z ) ∈ T n × R n × R 2 m , ε is a small parameter, and M ( ω ) 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.