Fractal characterization for subharmonic motion of completely inelastic bouncing ball

Abstract

The motion of a completely inelastic ball dropped vertically on the vibrating table will undergo a series of subharmonic bifurcations, controlled solely by the normalized vibration acceleration. It has been shown that the bifurcation diagram for the ball’s motion consists of almost equally spaced dense regions, in which the bifurcation behavior is sensitively dependent on the control parameter. The dense regions have complex interior geometrical structures. Here they are treated as fractal entities, and the fractal dimension for each of them is calculated. It is shown that the magnitude of the fractal dimension gradually increases, approaching a constant around 1.785.