The Dynamics of a Simplified Pin-ball Machine

In this paper we study the dynamics of an impact oscillator with a modified reset law derived from considering a problem (the pin-ball machine) with an active impact. Typical studies of the impact oscillator consider impacts which are governed by 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 pin-ball machine. In such a machine there exist bumpers which repel the pin-ball at high velocity when an (even slight) impact is made, imparting an additional velocity V to the rebounding pin-ball. Such impacts do not obey the normal laws and in this paper we study how to model them and the subtle dynamics which arises. The analysis proceeds by deriving a one-dimensional map which models the impacts. This map takes the form of a piece-wise linear/square-root map with a discontinuity of size proportional to V . The resulting map is similar in many aspects to a one-dimensional ’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 pin-ball machine.