  We study the dynamics of a particle in a horizontally and periodically shaken
box as a function of the box parameters and the coefficient of restitution. For
certain parameter values, the particle becomes regularly chattered at one of
the walls, thereby loosing all its kinetic energy relative to that wall. The
number of container oscillations between two chattering events depends in a
fractal manner on the parameters of the system. In contrast to a vertically
vibrated particle, for which chattering is claimed to be the generic fate, the
horizontally shaken particle can become trapped on a periodic orbit and follow
the period-doubling route to chaos when the coefficient of restitution is
changed. We also discuss the case of a completely elastic particle, and the
influence of friction between the particle and the bottom of the container.
