Relaxation Oscillations in a Nonsmooth Oscillator with Slow-Varying External Excitation

Abstract

The main purpose of the paper is to explore the influence of the coupling of two scales on the dynamics of a nonsmooth dynamical system. Based on a typical Chua’s circuit, by introducing a nonlinear resistor with piecewise characteristics as well as a harmonically changed electric source, a modified nonsmooth model is established, in which the coupling of two scales in frequency domain exists. Different types of bursting oscillations, appearing in the combination of large-amplitude oscillations, called spiking oscillations ([Formula: see text]), and small-amplitude oscillations or at rest, denoted by quiescent states ([Formula: see text]), can be observed with the variation of the exciting amplitude. When the exciting frequency is relatively small, by regarding the whole exciting term as a slow-varying parameter, the original system can be transformed into a generalized autonomous system. The phase space can be divided into three regions by the nonsmooth boundaries, in which the trajectory is governed by three different subsystems, respectively. Based on the analysis of the three subsystems as well as the behaviors on the nonsmooth boundaries, all the equilibrium branches and their bifurcations can be obtained, which can be employed to investigate the mechanism of the bursting oscillations. It is found that, for relatively small exciting amplitude, since no bifurcation on the equilibrium branches can be realized with the variation of the slow-varying parameter, the system behaves in periodic movement, which may evolve to bursting oscillations when a pair of fold bifurcations occurs with the increase of the exciting amplitude. Further increase of the exciting amplitude may lead to more complicated bursting oscillations, which may bifurcate into two coexisted asymmetric bursting attractors via symmetric breaking. Interaction between the two attractors may result in an enlarged symmetric bursting attractor, in which more forms of bifurcations at the transitions between the quiescent states and repetitive spiking states can be observed.