Dynamics of an impact oscillator near a degenerate graze

We give a complete analysis of low-velocity dynamics close to grazing for a generic one degree of freedom impact oscillator. This includes nondegenerate (quadratic) grazing and minimally degenerate (cubic) grazing, corresponding respectively to nondegenerate and degenerate chatter. We also describe the dynamics associated with generic one-parameter bifurcation at a more degenerate (quartic) graze, showing in particular how this gives rise to the often-observed highly convoluted structure in the stable manifolds of chattering orbits. The approach adopted is geometric, using methods from singularity theory.