The codimension-two bifurcation for the recent proposed SD oscillator

In this paper, the complicated nonlinear dynamics at the equilibria of SD oscillator, which exhibits both smooth and discontinuous dynamics depending on the value of a parameter α , are investigated. It is found that SD oscillator admits codimension-two bifurcation at the trivial equilibrium when α =1. The universal unfolding for the codimension-two bifurcation is also found to be equivalent to the damped SD oscillator with nonlinear viscous damping. Based on this equivalence between the universal unfolding and the damped system, the bifurcation diagram and the corresponding codimension-two bifurcation structures near the trivial equilibrium are obtained and presented for the damped SD oscillator as the perturbation parameters vary.