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A deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate. In the case of perfect elastic impacts, the equation reduces to the “standard mapping” which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells. It is shown that, for sufficiently large excitation velocities and a coefficient of restitution close to one, this deterministic dynamical system exhibits large families of irregular non-periodic solutions in addition to the expected harmonic and subharmonic motions. The physical significance of these and other chaotic motions which appear to occur frequently in non-linear oscillations is discussed.