Chaotic and Regular Motion in Dissipative Gravitational Billiards

Abstract

We consider the motion of a particle subjected to the constant gravitational field and scattered inelasticaly by hard boundaries which possess the shape of parabola, wedge, and hyperbola. The billiard itself performs oscillations. The linear dependence of the restitution coefficient on the particle velocity is assumed. We demonstrate that this dynamical system can be either regular or chaotic, which depends on the billiard shape and the oscillation frequency. The trajectory calculations are compared with the experimental data; a good agreement has been achieved. Moreover, the properties of the system has been studied by means of the Lyapunov exponents and the Kaplan-Yorke dimension. Chaotic and nonuniform patterns visible in the experimental data are interpreted as a result of large embedding dimension.