Degenerate orbit transitions in a one-dimensional inelastic particle system

Continuous transitions between different periodic orbits in a one-dimensional inelastic particle system are investigated. We show that continuous transitions that occur when adding or subtracting a single collision are, generically, of co-dimension 2. We give a full mechanical description of the system and explain why this is the case. Surprisingly, we also show that there are an infinite set of degenerate transitions of co-dimension 1. We provide a theoretical analysis that gives a simple criteria to classify which transitions are degenerate purely using the discrete set of collisions that occur in the orbits. Our analysis allows us to understand the nature of the degeneracy. We also show that higher degrees of degeneracy can occur, and provide an explanation. © 2010 The American Physical Society.