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Degenerate Grazing Bifurcations in a Simple Bilinear Oscillator

Degenerate grazing bifurcation occurring in a simple bilinear oscillator, namely the limit discontinuous case of the smooth and discontinuous (SD) oscillator, is investigated by numerical simulations. The unperturbed system has a saddle-like singularity at the origin with two periodic orbits grazing at the same point. The matrix in the leading-order truncation of the Poincare map at a grazing bifurcation for either of the two periodic orbits has a zero eigenvalue. Our numerical experiments suggest that the near-grazing dynamics are much more complicated than previously found. The results obtained in this paper is expected to be useful in developing analytical methods of this type of degenerate grazing bifurcations.