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One of the most interesting properties of an impacting system is the possibility of an infinite number of impacts occurring in a finite time (such as a ball bouncing to rest on a table). Such behaviour is usually called chatter. In this paper we make a systematic study of chattering behaviour for a periodically forced, single-degree-of-freedom impact oscillator with a restitution law for each impact. We show that chatter can occur for such systems and we compute the sets of initial data which always lead to chatter. We then show how these sets determine the intricate form of the domains of attraction for various types of asymptotic periodic motion. Finally, we deduce the existence of periodic motion which includes repeated chattering behaviour and show how this motion is related to certain types of chaotic behaviour.

We examine the change in behaviour of the solutions of a simple one degree of freedom, periodically forced, impact oscillator following a grazing bifurcation in which an impact of zero velocity occurs following a change in one of the parameters of the system. It is shown that such a bifurcation leads: to intermittent chaotic b�haviour with low velocity impacts followed by an irregular sequence of high velocity impacts. We also show that there is a natural, discontinuous oneMdimensional map associated with this relating one low velocity impact to the next and the properties of this map are analysed.We also construct the bifurcation diagram of the change in behaviour and show that this contains a series of periodic windows, with the period of the solutions increasing monotonicaUy by one in each successive window as the bifurcation point is approached.By restricting our attention to the resonant case where the forcing frequency is twice the natural frequency of the oscillator it is possible to make asymptotic estimates of the form of the intermittent chaotic behaviour and these estimates are compared with some numerical calculations.

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