Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach

We perform a bifurcation analysis of normal-internal resonances in parametrized families of quasi-periodically forced Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the 'backbone' system; forced, the system is a skew-product flow with a quasi-periodic driving with n basic frequencies. The dynamics of the forced system are simplified by averaging over the orbits of a linearization of the unforced system. The averaged system turns out to have the same structure as in the well-known case of periodic forcing (n = 1); for a real analytic system, the non-integrable part can even be made exponentially small in the forcing strength. We investigate the persistence and the bifurcations of quasi-periodic n-dimensional tori in the averaged system, filling normal-internal resonance 'gaps' that had been excluded in previous analyses. However, these gaps cannot completely be filled up: secondary resonance gaps appear, to which the averaging analysis can be applied again. This phenomenon of 'gaps within gaps' makes the quasi-periodic case more complicated than the periodic case.