  We have studied a dissipative version of a one-dimensional Fermi accelerator
model. The dynamics of the model is described in terms of a two-dimensional,
nonlinear area-contracting map. The dissipation is introduced via innelastic
collisions of the particle with the walls and we consider the dynamics in the
regime of high dissipation. For such a regime, the model exhibits a route to
chaos known as period doubling and we obtain a constant along the bifurcations
so called the Feigenbaum's number $\delta$.
