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Some dynamical properties for a dissipative time-dependent oval-shapedbilliard are studied. The system is described in terms of a four-dimensionalnonlinear mapping. Dissipation is introduced via inelastic collisions of theparticle with the boundary, thus implying that the particle has a fractionalloss of energy upon collision. The dissipation causes profound modifications inthe dynamics of the particle as well as in the phase space of the nondissipative system. In particular, inelastic collisions can be assumed as anefficient mechanism to suppress Fermi acceleration of the particle. Thedissipation also creates attractors in the system, including chaotic. We showthat a slightly modification of the intensity of the damping coefficient yieldsa drastic and sudden destruction of the chaotic attractor, thus leading thesystem to experience a boundary crisis. We have characterized such a boundarycrisis via a collision of the chaotic attractor with its own basin ofattraction and confirmed that inelastic collisions do indeed suppress Fermiacceleration in two-dimensional time dependent billiards.

Boundary crisis and suppression of Fermi acceleration in a dissipative two dimensional non-integrable time-dependent billiard