Collapse of periodic orbits in a driven inelastic particle system

The dynamical behavior of a one-dimensional inelastic particle system with particles of unequal mass traveling between two walls is investigated. The system is driven by adding energy at one of the walls while the other wall is stationary and does not add energy. By deriving analytic solutions for the periodic orbits of this system, we show that there are a countable infinity of critical mass ratios at which the particle dynamics become highly degenerate in the following sense. As the mass ratio passes through these critical points, large numbers of stable periodic orbits can collapse onto a single trivial orbit. We show that the widely studied equal-mass systems represent one of these critical points and are therefore such a degenerate case. We also show that in the elastic limit the number of orbits that collapse onto the single trivial orbit can become arbitrarily large.