INTERMITTENCY IN IMPACT OSCILLATORS CLOSE TO RESONANCE

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 behaviour with low velocity impacts followed by an irregular sequence of high velocity impacts. We also show that there is a natural, discontinuous one-dimensional 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 monotonically 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.