Effect of air damping on dynemical behaviors of a completely inelastic bouncing ball

A ball dropped on a vertically vibrating table exhibits intricate dynamical behaviors including period-doubling bifurcations and chaos. If the collision between the ball and the table is completely inelastic, the motion of the ball is always periodic, and the plateaus caused by saddle-node instability and clumping structures for periodic trajectories occur in the bifurcation diagram. Here the effect of air damping on the dynamics of the ball with zero elasticity is analyzed. The air damping is treated as linear viscous one. It is shown that a weak air damping does not change the sequence of bifurcations, but makes the bifurcation points shift to larger values and broadens the transverse dimensions of the plateaus and the clumping zones in the diagrams. However, when the air damping becomes larger, overlapping between the plateaus and clumping zones takes place. In the overlapping section, the mechanism originally leading to periodic motion is destroyed, and chaos is introduced.