Global dynamics and bifurcation for a discontinuous oscillator with irrational nonlinearity

The objective is to carry out thorough analysis of global dynamics and bifurcation for a discontinuous oscillator, showing the transition of dynamics between the piecewise smooth system (with irrational nonlinearity) and the piecewise linear system. The existence of one or three equilibrium sets is proven by the approach of Filippov. Based on non-smooth Lyapunov functions and their set-valued derivatives, asymptotical stability especially finite-time asymptotical stability is shown for the equilibrium sets. In particular, some conditions are established guaranteeing the uniqueness of an equilibrium set and its global finite-time asymptotical stability. Making use of the parametric representation, codimension-one bifurcation curves and a codimension-two bifurcation point are obtained. We give numerical simulations and bifurcation diagram to illustrate the transition of dynamics and verify the results. We further clarify some previous misconceptions in Li et al. (2016).