Degenerate grazing bifurcations in a three-degree-of-freedom impact oscillator

This paper presents the analysis of the degenerate grazing bifurcation in a three-degree-of-freedom impact oscillator by studying the bifurcations of near-grazing period-one impact motion near the degenerate grazing point. Actually, this paper extends the higher-order zero time discontinuity mapping to perform the perturbation analysis of characteristic equation of period-one impact motion and obtains feasible eigenvalue approximation to study the potential bifurcations. The shooting method is applied to verify the validity of the derived approximation and corresponding computation results. In addition to the known bifurcation scenarios of saddle-node and period-doubling, novel Neimark-Sacker bifurcation and related co-dimension two bifurcation points of near-grazing period-one impact motion are also found to arise near the degenerate grazing point in a three-degree-of-freedom impact oscillator. For the in-depth understanding of near-grazing dynamics, the obtained results are compared with the reported results in the single- and two-degree-of-freedom impact oscillators.