Lagrangian tori near resonances of near-integrable Hamiltonian systems

We study families of Lagrangian tori that appear in a neighbourhood of a resonance of a near-integrable Hamiltonian system. Such families disappear in the 'integrable' limit ε → 0. Dynamics on these tori are oscillatory in the direction of the resonance phases and rotating with respect to the other (non-resonant) phases. We also show that, if multiplicity of the resonance equals one, generically these tori occupy a set of a large relative measure in the resonant domains in the sense that the relative measure of the remaining 'chaotic' set is of the order $\sqrt \varepsilon$ . Therefore, for small ε > 0 a random initial condition in a $\sqrt \varepsilon$ -neighbourhood of a single resonance occurs inside this set (and therefore generates a quasi-periodic motion) with a probability much larger than in the 'chaotic' set. We present results of numerical simulations and discuss the form of projection of such tori to the action space. At the end of section 4 we discuss the relationship of our results and a conjecture that tori (in a near-integrable Hamiltonian systems) occupy all the phase space except a set of measure ~ε.