Bursting oscillations with non-smooth Hopf-fold bifurcations of boundary equilibria in a piecewise-smooth system

This paper constructs a slow–fast dynamical system by using a fourth-order Chua’s circuit to study the dynamics of bursting oscillations in non-smooth systems. The considered system is composed of three smooth fast-slow subsystems connected by two switching manifolds. As the bifurcation behaviors of the fast subsystem are analyzed, a combination of theoretical analysis and numerical simulation methods is utilized. In the smooth fast subsystem, subcritical Hopf bifurcations are obtained. The bifurcation analysis of the non-smooth fast subsystem is divided into two parts: one is the bifurcation analysis of the limit cycles and chaotic attractors, in which the non-smooth fold cycle bifurcations, period-doubling bifurcations of limit cycles, and bifurcations of chaotic attractors are obtained; the other is the bifurcation analysis of the boundary equilibria, in which a kind of multiple crossing bifurcation, exhibiting the behaviors of both Hopf and fold bifurcations, termed “non-smooth Hopf-fold bifurcation”, is observed. On the bases of bifurcation analysis, the dynamics of five typical bursting is uncovered. Study suggests that the slow passage effect through the subcritical Hopf bifurcation can create the non-smooth hysteresis loop of slow state variable. The delayed transition to spiking area enables occurrence of different busting patterns. The non-smooth Hopf-fold bifurcation point provides not only the transition point from a resting state to a spiking state, but also the spiking area in the bursting oscillations.