Non-smooth bifurcations on the bursting oscillations in a dynamic system with two timescales

Abstract

The main purpose of the paper was to explore the structures of bursting oscillations as well as the mechanism of non-smooth dynamic system with two timescales. Based on the typical Chua’s system, a non-smooth dynamic system with two timescales is established, which can be considered as the coupling of slow and fast subsystems. Possible bifurcations as well as the critical conditions related to the equilibrium points of the fast subsystem in different regions divided by the switching boundaries are derived. The distribution of the eigenvalues related to the generalized Jacobian matrix reveals that multiple crossing bifurcations may occur at the switching boundaries, which may influence the dynamics of system. Periodic bursting oscillations can be observed, in which the fold-type non-smooth bifurcations connect the quiescent state and spiking state, leading to the symmetric focus/focus-fold/fold bursting attractors. Different behaviors can be observed at the intersection points between the trajectories and the switching boundaries, which can be explained by the characteristics of the equilibrium points in two regions beside the boundaries. Furthermore, it is found that the real parts of the pair of complex conjugate eigenvalues related to the foci may dramatically depend on certain parameters, which is employed to account for the change in the oscillation patterns related to the spiking state.