A Quasi-Periodic Poincaré's Theorem

Abstract

We study the persistence of invariant tori on resonant surfaces of a nearly integrable Hamiltonian system under the usual Kolmogorov non-degenerate condition. By introducing a quasi-linear iterative scheme to deal with small divisors, we generalize the Poincaré theorem on the maximal resonance case (i.e., the periodic case) to the general resonance case (i.e., the quasi-periodic case) by showing the persistence of majority of invariant tori associated to non-degenerate relative equilibria on any resonant surface.