Subharmonic and grazing bifurcations for a simple bilinear oscillator

• The limit case of the SD oscillator is studied for subharmonic orbits.
• The existence, stability and bifurcations of the subharmonic orbits are considered.
• Grazing bifurcations are discussed and the bifurcations are likely discontinuous.
• Effects of the size of dissipation on bifurcations are investigated.

In this paper, subharmonic and grazing bifurcations for a simple bilinear oscillator, namely the limit discontinuous case of the smooth and discontinuous (SD) oscillator are studied. This system is an important model that can be used to investigate the transition from smooth to discontinuous dynamics. A combination of analytical and numerical methods is used to investigate the existence, stability and bifurcations of symmetric and asymmetric subharmonic orbits. Grazing bifurcations for a particular periodic orbit are also discussed and numerical results suggest that the bifurcations are discontinuous. We show via concrete numerical experiments that the dynamics of the system for the case of large dissipation is quite different from that for the case of small dissipation.