The dynamics of a simplified pinball machine

In this paper, we study the dynamics of an impact oscillator with a modified reset law derived from considering a problem (the pinball machine) with an ‘active impact’. Typical studies of the impact oscillator consider impacts that are governedby Newton's law of restitution where the velocity after impact is r times less than the incoming velocity. But in this paper, we consider an active impact modelling impacts, which occur in a pinball machine. In such a machine, there exist bumpers that repel the pinball at high velocity when an (even slight) impact is made, imparting an additional velocity V to the rebounding pinball. Such impacts do not obey the normal laws and in this paper, we study how to model them and the subtle dynamics that arises. The analysis proceeds by deriving a 1D map that models the impacts. This map takes the form of a piecewise linear/square-root map with a discontinuity of size proportional to V. The resulting map is similar in many aspects to a 1D ‘map-with-a-gap’ but also inherits features of the square-root map. We show how the subtle interplay between these two maps leads to the creation of a very large number of new period orbits, which might explain some of the complexity observed in the dynamics of a true pinball machine.