| When N-beads slide along a frictionless hoop, their collision sequence gives rise to a dynamical system that can be studied via matrix products. The problem is formulated for general N and some basic theorems are stated regarding the eigenvalues of the collision matrices and their products. The three-bead case is studied in detail. Each collision sequence can be viewed as a billiard trajectory in a right triangle with non-standard reflection rules. Both periodic and dense collision sequences are discussed, and a series of conjectures based on computational evidence are formulated. Dense trajectories generated both from a chaotic collision sequence and a randomly ordered matrix sequence are compared via the eigenvalue distributions and autocorrelation matrices they produce.; The problem of three beads of masses m, m-epsilon, m+epsilon is studied for epsilon+/-0. When epsilon=0, the spectrum is discrete on the unit circle underlying the fact that the dynamics are regular. For epsilon>0, the eigenvalue spectrum produced by a deterministically chaotic trajectory is compared to spectra produced by two different stochastic problems, a randomly generated sequence of matrix products and a random walk process on the unit circle. We describe how to use the chaotic collision sequences as the basis for a random number generating algorithm. By examining both the runs and reverse arrangement tests, we conclude that the degree of randomness produced by these sequences is equivalent to Matlab's rand() routine.; Finally, the case where the masses are scaled so that m1=1/epsilon, m2=1, m3=1-epsilon, for 0+/-epsilon+/-1 is investigated. The singular limits epsilon=0 and epsilon=1 correspond to two equal mass beads colliding on the ring with and without a wall respectively. In both cases, all solutions are periodic, and the eigenvalue distributions associated with products of collision matrices are discrete. The regime, parametrically connecting these two states (0<epsilon<1), is examined, and the eigenvalue distribution is shown to be generically uniform around the unit circle, implying that the dynamics are no longer periodic. We characterize how the uniform spectrum collapses from continuous to discrete in the two singular limits. |
