The evolution mechanism of different forms of bursting oscillations in non-smooth dynamical systems

Abstract

The dynamics of the systems with non-smoothness and the coupling of multiple scales have become the key subject because of their wide applications in science and engineering problems. The main purpose of the manuscript is to investigate the complicated dynamics in non-smooth systems caused by the coupling of two scales in frequency domain. Based on a common model of the boost converter controlled by a SMC-Washout as an example, By introducing a periodically changed electric power source and taking suitable parameter values, a piecewise non-smooth Filippov system with two scales in frequency domain is established. For the case when the exciting frequency is far less than the natural frequency, the whole exciting term can be regarded as a slow-varying parameter. Upon the analysis of the two subsystems located in two regions divided by the non-smooth boundary, the evolution of the equilibrium branches as well as the bifurcations with the variation of the slow-varying parameter are derived. Two typical cases corresponding to different distribution of the equilibrium branches as well as the bifurcations with the variation of the slow-varying parameter are taken into consideration, in which different types of bursting oscillations can be observed. By introducing the concept of the transformed phase portrait, the associated bifurcation mechanism of the bursting oscillations is presented. It is found that, the equilibrium branches and the related bifurcations may change with the variation of parameters, which may not only affect the quiescent states and the spiking states, but also influence the bifurcations between the two states, leading to different forms of bursting oscillations. It is pointed out that, when the trajectory of system jumps between different stable equilibrium branches of the two subsystems located in two associated regions, the types of point/point bursting oscillations may be obtained, while with the variation of the parameters, when the stable limit cycles involves the attractors, the types of point/cycle bursting oscillations may appear.