Lagrangian tori near resonances of near-integrable Hamiltonian systems

Abstract

In this paper we study families of Lagrangian tori that appear in a neighborhood of a resonance of a near-integrable Hamiltonian system. Such families disappear in the "integrable" limit $\varepsilon\to 0$. Dynamics on these tori is 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 large relative measure in the resonant domains in the sense that the relative measure of the remaining "chaotic" set is of order $\sqrt\varepsilon$. Therefore for small $\varepsilon > 0$ a random initial condition in a $\sqrt\varepsilon$-neighborhood 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.