Bursting oscillations with multiple crossing bifurcations in a piecewise-smooth system

Abstract

This paper aims to investigate the non-smooth bifurcations and to uncover the underlying dynamics that lead to bursting patterns within a two-scale piecewise-smooth system. The system is established by applying a modification scheme to a fourth-order Chua’s circuit, with a periodic external excitation current acting as the slow state variable. Several smooth as well as non-smooth bifurcations are discovered within the fast subsystem by utilizing theoretical and numerical methods. Two special non-smooth bifurcations have been discussed. The first is the multiple crossing bifurcation involving the boundary equilibrium, which exhibits the behavior of both the turning point and Hopf bifurcation. The second arises from an encounter between a saddle-focus and the trajectory of a non-smooth chaotic solution, which can result in the vanishing or appearance of a non-smooth chaotic attractor. Four typical bursting patterns associated with these two non-smooth bifurcations in the established slow-fast system are observed, and the mechanisms behind them are revealed based on bifurcation analysis.