RESUMO
We characterize a transition from normal to ballistic diffusion in a bouncing ball dynamics. The system is composed of a particle, or an ensemble of noninteracting particles, experiencing elastic collisions with a heavy and periodically moving wall under the influence of a constant gravitational field. The dynamics lead to a mixed phase space where chaotic orbits have a free path to move along the velocity axis, presenting a normal diffusion behavior. Depending on the control parameter, one can observe the presence of featured resonances, known as accelerator modes, that lead to a ballistic growth of velocity. Through statistical and numerical analysis of the velocity of the particle, we are able to characterize a transition between the two regimes, where transport properties were used to characterize the scenario of the ballistic regime. Also, in an analysis of the probability of an orbit to reach an accelerator mode as a function of the velocity, we observe a competition between the normal and ballistic transport in the midrange velocity.
RESUMO
Physical systems such as optical traps and microwave cavities are realistically modeled by billiards with soft walls. In order to investigate the influence of the wall softness on the billiard dynamics, we study numerically a smooth two-dimensional potential well that has the elliptical (hard-wall) billiard as a limiting case. Considering two parameters, the eccentricity of the elliptical equipotential curves and the wall hardness, which defines the steepness of the well, we show that (1) whereas the hard-wall limit is integrable and thus completely regular, the soft wall elliptical billiard exhibits chaos, (2) the chaotic fraction of the phase space depends nonmonotonically on the hardness of the wall, and (3) the effect of the hardness on the dynamics depends strongly on the eccentricity of the billiard. We further show that the limaçon billiard can exhibit enhanced chaos induced by wall softness, which suggests that our findings generalize to quasi-integrable systems.
RESUMO
The ballistic increase for the velocity of a particle in a bouncing-ball model was investigated. The phenomenon is caused by accelerating structures in phase space known as accelerator modes. They lead to a regular and monotonic increase of the velocity. Here, both regular and ballistic Fermi acceleration coexist in the dynamics, leading the dynamics to two different growth regimes. We characterized deaccelerator modes in the dynamics, corresponding to unstable points in the antisymmetric position of the accelerator modes. In control parameter space, parameter sets for which these accelerations and deaccelerations constitute structures were obtained analytically. Since the mapping is not symplectic, we found fractal basins of influence for acceleration and deacceleration bounded by the stable and unstable manifolds, where the basins affect globally the average velocity of the system.
RESUMO
Some phase space transport properties for a conservative bouncer model are studied. The dynamics of the model is described by using a two-dimensional measure preserving mapping for the variables' velocity and time. The system is characterized by a control parameter ε and experiences a transition from integrable (ε=0) to nonintegrable (ε≠0). For small values of ε, the phase space shows a mixed structure where periodic islands, chaotic seas, and invariant tori coexist. As the parameter ε increases and reaches a critical value εc, all invariant tori are destroyed and the chaotic sea spreads over the phase space, leading the particle to diffuse in velocity and experience Fermi acceleration (unlimited energy growth). During the dynamics the particle can be temporarily trapped near periodic and stable regions. We use the finite time Lyapunov exponent to visualize this effect. The survival probability was used to obtain some of the transport properties in the phase space. For large ε, the survival probability decays exponentially when it turns into a slower decay as the control parameter ε is reduced. The slower decay is related to trapping dynamics, slowing the Fermi Acceleration, i.e., unbounded growth of the velocity.