Análise de escala em bilhares com fronteiras móveis

Abstract

We study numerically the scaling properties of some dynamical systems. Near the transition from the integrable to the non-integrable regime of complete e simplified versions of Fermi-Ulam model, we investigate the region of low energy (chaotic sea). We evaluate average quantities as functions (a) of the iteration number n or the time t, (b) of the initial velocity and (c) of the control parameter. We also investigate the scaling properties of the simplified bouncer model by mapping it in the standard model. We obtain the scaling properties (i) of the integrable to non-integrable transition (weakly non-linear regime), (ii) of the transition from the regime of limited energy growth to the regime of Fermi acceleration (unlimited energy growth) and (iii) for the regime of big values of the non-linearity parameter. We also study the properties of the bouncer model with inelastic collisions between the particle and the wall. We obtain the scaling description of the transition from the unlimited to the limited energy growth when the dissipation is introduced. We describe some properties of the phase space of the pulsating circular billiard and we obtain the scaling description of the chaotic sea nearthe integrable to the non-integrable transition. Finally we study a hybridversion of the of Fermi-Ulam and bouncer models. Regarding inelastic collisions we present some properties of the phase space, as occurrence of crisis and cascades of period doubling.