Multiple island chains in wave-particle interactions

We analyze the isochronous island chains that appear in the Poincaré sections of near integrable twist systems. When the system presents just one resonant perturbation with a winding number, the number of chains is constant and it is completely determined by the perturbation. However, for systems that are perturbed by an infinite number of resonant perturbations with the same winding number, the number of isochronous chains depends on the superposition of the perturbations and it is a function of the parameters. Considering a system that describes wave-particle interaction, we show that the number of island chains increases without limit when the wave period or wave number are increased.