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We study the dynamic evolution of a bouncing ball on a vibrating platform, such as the membrane of a loudspeaker, as a function of its coefficient of restitution and demonstrate primarily that the presence of chaos in this system is by no means inevitable and its development is by no means obvious. Indeed, we show that generic trajectories starting under experimental conditions terminate in a region of ``chattering'' (where the memory of earlier dynamics is lost), before repeating themselves in a periodic way. As a consequence, except possibly for the strictly elastic case (=1), the evolution to chaos via a period-doubling route will not be observed. Our arguments are corroborated by numerical studies, concerning especially the divergence of the mean period of generic trajectories as the elastic limit is approached (1).
Luck et al. (Mon,) studied this question.