Stability of periodic modes and bifurcation behaviors in a bouncing-dimer system

Recent experiments and simulations have shown that a dumbbell-shaped body, termed as a dimer, on a vibrating plate exhibits an amazing self-ordered phenomenon, in which its horizontal motion can take directed transport behavior when the body bounces periodically. While the existing investigations have detailed its dynamics comprehensively, it still remains unclear how the physical parameters of the system affect the emergence of the intriguing phenomenon. In this paper, we first reduce the numerical model of the system into a simple model under an assumption that one end of the body always stays on the vibrating plate, while the other end bounces periodically. The simplification on modeling enables us to establish a discrete map focusing on the impact of the bouncing end. Then, the stability of the periodic behavior can be addressed by analyzing the property of the impact map around its fixed points. Finally, the stability properties of the self-ordered behaviors can be quantified by an explicit relation between the coefficient of restitution e and the vibration intensity Γ. The developed theoretical results demonstrate that the system exhibits rich nonlinear phenomena, including simple periodic modes, chattering, multiperiodicity, period-doubling bifurcation, and chaos. Guided by the theoretical predictions, we performed investigations via experiments and simulations. Comparison with the results obtained from the numerical model and the experiments reveals that the analytical results are very effective in accurate predictions.