A simplified Fermi Accelerator Model under quadratic frictional force

some dynamical properties for a simplified version of a one-dimensional fermi accelerator model under the action of a small dissipation is studied. the dissipation is introduced via a damping force which is assumed to be proportional to the square particle's velocity. the dynamics of the model is described by using a two-dimensional, nonlinear area contracting mapping for the variables velocity of the particle and time. our results confirm that the structure of the phase space of the conservative version is replaced by a large number of attracting periodic orbits. for a fixed set of control parameters, we obtain many periodic attractors and show that most of them posses low period. the stable orbits produce a complex structure of basin of attraction whose limit cover almost all phase space, thus suggesting a fractality.