Resonant structure of integrable and near-integrable two-dimensional systems

The structure of the resonances in two-dimensional (2D) Hamiltonian systems is analyzed in the frequency space. We define the frequency-frequency curves (FFC’s), discuss some of their mathematical characteristics, and with that basis in mind, we classify 2D Hamiltonian systems from the point of view of the shape of their FFC’s. We discuss this classification through a number of prototypical examples. The analytical conditions allow us to predict relevant features of the resonances’ structure of integrable systems. We also consider under which conditions the FFC picture persists (in a coarse-grained sense) for near-integrable systems. This allows us to qualitatively understand and quantitatively predict the energy of the transition to global stochasticity by using Chirikov’s criterion for the overlapping of resonances. These ideas are applied to Sinai’s billiard with two different perturbations.