Global properties of generic real–analytic nearly–integrable Hamiltonian systems

We introduce a new class Gns of generic real analytic potentials on Tn and study global analytic properties of natural nearly-integrable Hamiltonians 12|y|2 + epsilon f (x), with potential f is an element of Gns, on the phase space M = B x Tn with B a given ball in Rn. The phase space M can be covered by three sets: a 'non-resonant' set, which is filled up to an exponentially small set of measure e-cK (where K is the maximal size of resonances considered) by primary maximal KAM tori; a 'simply resonant set' of measure root epsilon Ka and a third set of measure epsilon Kb which is 'non perturbative', in the sense that the H-dynamics on it can be described by a natural system which is not nearly-integrable. We then focus on the simply resonant set - the dynamics of which is particularly interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori) - and show that on such a set the secular (averaged) 1 degree-of-freedom Hamiltonians (labeled by the resonance index k is an element of Zn) can be put into a universal form (which we call 'Generic Standard Form'), whose main analytic properties are controlled by only one parameter, which is uniform in the resonance label k. (c) 2023 Elsevier Inc. All rights reserved.