As an example of a nonperiodically forced system, the dynamics of the Fermi accelerator (ball bouncing on piston oscillating periodically in time) subject to viscous friction is studied numerically and analytically. Its evolution is determined from a two-dimensional (2D) map, realized as two coupled difference equations. Dimensional reduction is achieved analytically in a simple high-dissipation approximation yielding a version of the circle map. Numerical experiments using this map compare quantitatively and qualitatively well with the exact 2D map in the high- and medium-dissipation regimes, respectively. Quasiperiodic, periodic, and chaotic behavior are observed in the region around the critical line of the circle map in accordance with the Arnol’d tongues scheme. For higher values of the control parameter, an infinite number of ‘‘chaos-period doubling-crisis’’ sequences characterize the bifurcation diagram of the system. The mechanisms for the sudden death and expansion of chaotic attractors are easily understood via return maps of the circle. The dynamics of the weakly damped Fermi accelerator is qualitatively well described by a simplified map that is a nonstandard dissipative 2D map with a nonconstant Jacobian. The possibility of the damped Fermi accelerator not obeying universal properties of a certain class of dissipative systems is discussed.

• Received 20 April 1990

DOI:https://doi.org/10.1103/PhysRevA.42.7155