Two balls in one dimension with gravity

A study is reported of a simple dynamical system with two degrees of freedom having discontinuities due to collisions. It consists of two point masses or balls constrained to move in one dimension above a floor in a constant gravitational field. All collisions are assumed to be elastic. When the ratio r of the upper mass to the lower mass is less than unity the motion is chaotic almost everywhere. On the other hand, when r>1 the motion shows typical Kolmogorov-Arnol’d-Moser behavior with quasiperiodic and chaotic trajectories coexisting in the phase space. It is shown that for particular values of the mass ratio, denoted by rn, a sequence of n rapid ball-ball collisions close to the floor has the net effect of reversing the velocities of both balls. This demonstration leads to the identification of families of stable and unstable fixed points of the Poincaré section, which to a considerable extent determine the overall structure of the map. By means of a method due to Lorenz, the largest Lyapunov exponent λ1 has been calculated for many values of the mass ratio and for a variety of trajectories. For chaotic trajectories, a plot of λ1 as a function of r is found to have local minima at the values rn corresponding to velocity-reversing collision sequences. This is thought to result from the fact that when r=rn the chaotic trajectories lie in many isolated regions of the phase space, whereas when r is different from any of the rn, the chaotic regions merge to form a single region of global chaos.