Continuous and discontinuous grazing bifurcations in impacting oscillators

This paper seeks to formulate conditions for the persistence of a local attractor in the immediate vicinity of periodic and quasiperiodic grazing trajectories in an impacting mechanical system. A local analysis based on the discontinuity-mapping approach is employed to derive a normal-form description of the dynamics near the grazing trajectory. In agreement with previous studies of grazing periodic trajectories, it is found that the catastrophic loss of a local attractor and strong instability characteristic of grazing bifurcations is directly associated with the repeated application of a square-root term that appears to lowest order in the normal-form expansion. Specifically, it is found that the square-root term is absent in the description of the dynamics normal to a quasiperiodic trajectory covering a co-dimension-one invariant torus resulting in a piecewise linear description of the normal dynamics and, at most, a weak instability. In contrast, for co-dimension-two or higher, the square-root term is generically present in the normal dynamics. Here, however, the quasiperiodicity of the grazing motion implies that there is no upper limit to the time between impacts on nearby trajectories suggesting the persistence of a local attractor for some interval about the parameter value corresponding to grazing. The results of the analysis are illustrated through a series of model examples.