Subharmonic motion of granular particles under vertical vibrations

We investigate the impact of granular particles at the bottom of the container, which is vibrating harmonically in the vertical direction. Experiments demonstrate that the impact strength, controlled by the normalized vibration acceleration, undergoes a series of subharmonic bifurcations, in the sequence of period-doubling, period-quadrupling, chaos, period-tripling, period-sextupling, chaos, period-quadrupling, period-octupling, and so on. It is observed that accompanying the impact bifurcations there always exists a condensed state of particles at the lower part of the bed. By using the completely inelastic bouncing-ball model, a possible explanation for the observed bifurcation phenomena is presented.