Amplitude effects on stepwise bifurcation in a bouncing ball system

In this work, we consider both experimentally and numerically a bouncing-ball system comprising a ball bouncing on a table that oscillates in a sinusoidal manner. We analyze the stepwise bifurcations observed in this system through which the maximum bounce height of the ball increases in a stepwise manner as the table oscillation frequency is increased. The experiments were conducted for multiple vibration amplitudes, and we examined the changes in the ball's motion as the table frequency was varied discretely. The results show that the transitions between motion states occur in a stepwise manner with respect to the driving frequency, and that the bifurcation points obtained numerically and experimentally exhibit close quantitative agreement. Parameterizing the system in terms of the acceleration amplitude of the table, the bifurcation structure for different amplitudes collapses onto a single curve, indicating that the onset and progression of bifurcations are influenced by the acceleration amplitude rather than by the oscillation frequency alone. A phase-space analysis considering Poincaré maps further reveals that each bifurcation introduces an additional state region; we find that the experimentally observed attractors appear more connected than their numerical counterparts due to the embedding reconstruction. Events associated with sticking regimes were observed in simulations but not in experiments, suggesting that small perturbations and measurement uncertainties may be sufficient to suppress such idealized behaviors.