Bursting oscillations as well as the mechanism with codimension-1 non-smooth bifurcation

Abstract

The coupling of different scales in nonlinear systems may lead to some special dynamical phenomena, which always behaves in the combination between large-amplitude oscillations and small-amplitude oscillations, namely bursting oscillations. Up to now, most of therelevant reports have focused on the smooth dynamical systems. However, the coupling of different scales in non-smooth systems may lead to more complicated forms of bursting oscillations because of the existences of different types of non-conventional bifurcations in non-smooth systems. The main purpose of the paper is to explore the coupling effects of multiple scales in non-smooth dynamical systems with non-conventional bifurcations which may occur at the non-smooth boundaries. According to the typical generalized Chua's electrical circuit which contains two non-smooth boundaries, we establish a four-dimensional piecewise-linear dynamical model with different scales in frequency domain. In the model, we introduce a periodically changed current source as well as a capacity for controlling. We select suitable parameter values such that an order gap exists between the exciting frequency and the natural frequency. The state space is divided into several regions in which different types of equilibrium points of the fast sub-system can be observed. By employing the generalized Clarke derivative, different forms of non-smooth bifurcations as well as the conditions are derived when the trajectory passes across the non-smooth boundaries. The case of codimension-1 non-conventional bifurcation is taken for example to investigate the effects of multiple scales on the dynamics of the system. Periodic bursting oscillations can be observed in which codimension-1 bifurcation causes the transitions between the quiescent states and the spiking states. The structure analysis of the attractor points out that the trajectory can be divided into three segments located in different regions. The theoretical period of the movement as well as the amplitudes of the spiking oscillations is derived accordingly, which agrees well with the numerical result. Based on the envelope analysis, the mechanism of the bursting oscillations is presented, which reveals the characteristics of the quiescent states and the repetitive spiking oscillations. Furthermore, unlike the fold bifurcations which may lead to jumping phenomena between two different equilibrium points of the system, the non-smooth fold bifurcation may cause the jumping phenomenon between two equilibrium points located in two regions divided by the non-smooth boundaries. When the trajectory of the system passes across the non-smooth boundaries, non-smooth fold bifurcations may cause the system to tend to different equilibrium points, corresponding to the transitions between quiescent states and spiking states, which may lead to the bursting oscillations.