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Nonlinear dynamics of a bouncing ball moving vertically in a gravitationalfield and colliding with a moving limiter is considered and the Poincare map,describing evolution from an impact to the next impact, is described.Displacement of the limiter is assumed as periodic, cubic function of time. Dueto simplicity of this function analytical computations are possible. Severaldynamical modes, such as fixed points, 2 - cycles and chaotic bands are studiedanalytically and numerically. It is shown that chaotic bands are created fromfixed points after first period doubling in a corner-type bifurcation. Equationfor the time of the next impact is solved exactly for the case of twosubsequent impacts occurring in the same period of limiter's motion makinganalysis of chattering possible.

Simple model of bouncing ball dynamics. Displacement of the limiter assumed as a cubic function of time