Dynamics and scaling properties for a one-dimensional impact system with two periodically vibrating walls

Abstract

We investigate the dynamics of a system composed of a particle suffering impacts between two heavy periodically vibrating walls. An original, nonlinear area preserving mapping is obtained. The control parameters of amplitude of perturbation and frequency of oscillation play an important role in the phase space, shaping the portion of chaotic seas, position of invariant curves and the amount of KAM islands. The study of the behavior of the root mean square velocity was made via analytical description and numerical simulations. We proposed scaling arguments to describe its dynamics and our results show remarkably good agreement between the theory and the simulations concerning a scaling invariance with respect to the control parameters. Also, an analysis of the diffusion coefficient confirms the validity of the scaling invariance, giving robustness to our modeling.