Nordmark map and the problem of large-amplitude chaos in impact oscillators

Abstract

Physical experiments have long revealed that impact oscillators commonly exhibit large-amplitude chaos over a narrow band of parameter values close to grazing bifurcations. This phenomenon is not explained by the square-root singularity of the Nordmark map, which captures the local dynamics to leading order, because this map does not exhibit such dynamics. In this paper, we compare a Poincaré map for a prototypical impact oscillator model with the corresponding Nordmark map. Though the maps agree to leading order, the Poincaré map exhibits a large-amplitude chaotic attractor while the Nordmark map does not because part of the attractor resides in a region of phase space where the two maps differ significantly.