Grazing in Impact Oscillators

Abstract

In the last talk on impact oscillators we reviewed some of their general behaviour and looked at the simple smooth bifurcations that occur when studying the stability of the simple (1, n) orbits. We now proceed to investigate the bifurcations which can occur when we consider discontinuous changes in behaviour. These are associated with grazing impacts of the trajectory of the particle with the obstacle. A graze leads to quite remarkable behaviour including an immediate bifurcation from a periodic orbit to a chaotic one. Such behaviour is unique to impact oscillators although [Nordmark] has shown that some of the behaviour associated with the discontinuities in the impact oscillator can be viewed as a limit of smooth bifurcations occuring in systems with smooth models for the impacts.