  Some dynamical properties for a dissipative time-dependent oval-shaped
billiard are studied. The system is described in terms of a four-dimensional
nonlinear mapping. Dissipation is introduced via inelastic collisions of the
particle with the boundary, thus implying that the particle has a fractional
loss of energy upon collision. The dissipation causes profound modifications in
the dynamics of the particle as well as in the phase space of the non
dissipative system. In particular, inelastic collisions can be assumed as an
efficient mechanism to suppress Fermi acceleration of the particle. The
dissipation also creates attractors in the system, including chaotic. We show
that a slightly modification of the intensity of the damping coefficient yields
a drastic and sudden destruction of the chaotic attractor, thus leading the
system to experience a boundary crisis. We have characterized such a boundary
crisis via a collision of the chaotic attractor with its own basin of
attraction and confirmed that inelastic collisions do indeed suppress Fermi
acceleration in two-dimensional time dependent billiards.
