DETERMINISTIC ANALYSIS OF THE PROBABILITY MACHINE

Nonlinear iterated mapping is applied to determine the trajectory of a ball falling through a lattice of horizontal pins (a pinball or Galto n machine). individual impacts between the ball and pins are described in a fully deterministic manner, accounting for rolling and sliding a s well as for repeated inelastic collisions, which do or do not involv e sliding. The basins of attraction for the different final outcomes a re obtained, and their distribution in the space of initial conditions is discussed. The system has certain properties in common with chaoti c systems. However, the ball experiences a finite number of collisions , and therefore the outcome is fully predictable. The apparent randomn ess arises through amplification of inaccuracies in the initial condit ions. By simulating the trajectories of a large number of balls with s lightly different initial conditions, the outcome distribution is dete rmined as a function of such parameters as the coefficient of restitut ion and the spacing of the pins to the size of the balls. A binomial d istribution is only obtained for a very specific set of parameters.