A theoretical characterization of scaling properties in a bouncing ball system

Analytical arguments are used to describe the behavior of the average velocity in the problem of an ensemble of particles bouncing a heavy and periodically moving platform. The dynamics of the system is described by using a two-dimensional mapping for the variables’ velocity and discrete time n. In the absence of dissipation and depending on the control parameter and initial conditions, diffusion in energy is observed. Considering the introduction of dissipation via inelastic collisions, we prove that the diffusion is interrupted and a transition from unlimited to limited energy growth is characterized. Our result is general and can be used when the initial condition is a very low velocity leading to a growth of average velocity with n or for large initial velocity where an exponential decay of the average velocity is observed. The results obtained generalize the scaling observed in the bouncer model as well as the stochastic and dissipative Fermi–Ulam model. The formalism can be extended to many other different types of models, including a class of time-dependent billiards.