Analytical determination for degenerate grazing bifurcation points in the single-degree-of-freedom impact oscillator

Abstract

Co-dimension-one grazing bifurcations of 1 / n impact periodic motions are often accompanied by saddle-node bifurcations or period-doubling bifurcations. The presence of certain degenerate grazing bifurcation points plays an important role in transitions between these two different co-dimension-one grazing bifurcation scenarios. When the saddle-node bifurcation line and period-doubling bifurcation line meet at a grazing bifurcation point, a degenerate grazing bifurcation occurs. By considering the existence and stability of the 1 / n impact periodic motions near grazing points in single-degree-of-freedom impact oscillators, an analytical method is developed with the aid of the discontinuity mapping technique to determine the certain degenerate grazing bifurcation points of 1 / n motions. It is found that the linear term in the local discontinuity mapping does not affect the distribution of certain degenerate grazing points. The unfolding in the neighborhood of degenerate grazing bifurcation points is verified by numerical simulations.