  Some dynamical properties of a bouncing ball model under the presence of an
external force modeled by two nonlinear terms are studied. The description of
the model is made by use of a two dimensional nonlinear measure preserving map
on the variables velocity of the particle and time. We show that raising the
straight of a control parameter which controls one of the nonlinearities, the
positive Lyapunov exponent decreases in the average and suffers abrupt changes.
We also show that for a specific range of control parameters, the model
exhibits the phenomenon of Fermi acceleration. The explanation of both
behaviours is given in terms of the shape of the external force and due to a
discontinuity of the moving wall's velocity.
