Bursting oscillations and mechanism of sliding movement in piecewise Filippov system

Abstract

Since the wide applications in science and engineering, the dynamics of non-smooth system has become one of the key research subjects. Furthermore, the interaction between different scales may result in special movement which can be usually described by the combination of large-amplitude oscillation and small-amplitude one. The influence of multiple scale on the dynamics of non-smooth system has received much attention recently. In this work, we try to explore the bursting oscillations and the mechanism of non-smooth Filippov system coupled by different scales in the frequency domain. Taking the typical periodically excited Duffing's oscillator for example a Filippov system coupled by two scales in the frequency domain is established when the difference in order between the excited frequency and the system natural frequency is obtained by introducing the piecewise control into the state variable and choosing suitable parameters. For the case in which the exciting frequency is far less than the natural frequency, the whole exciting term can be considered as a slow-varying parameter, also called slow subsystem, which leads to a generalized autonomous system, i.e., the fast subsystem. The equilibrium branches and the bifurcations of the fast subsystem along with the variation of the slow-varying parameter in different regions divided according to non-smooth boundary, can be derived. Two typical cases are taken into consideration, in which different distributions of the equilibrium branches and the relevant bifurcations of the fast subsystem may exist. It is pointed out that the variations of the parameters may influence not only the properties of the equilibrium branches, but also the structures of the bursting attractors. Furthermore, since the governing equation alternates between two subsystems located in different regions when the trajectory passes across the non-smooth boundary, the sliding movement along the non-smooth boundary of the trajectory can be observed under the condition of certain parameters. By employing the transformed phase portrait which describes the relationship between the state variable and the slow-varying parameter, the mechanisms of different bursting oscillations and sliding movements are investigated. The results show that bursting oscillations may exist in a non-smooth Filippov system coupled by two scales in the frequency domain. The alternations of the governing equation between different subsystems located in the two neighboring regions along the non-smooth boundary may result in a sliding movement of the trajectory along the non-smooth boundary.