The Research On Two Scale Effects And Bifurcation Mechanisms In Filippov Systems

The dynamics of the systems with non-smooth and the coupling of multiple scales have become one of the key subjects in the fields of nonlinear dynamics and control because of their wide applications in science and engineering.Since the mechanisms of many phenomena in such systems can not be expounded by the traditional nonlinear theories,such as the non-conventional bifurcations with the trajectory crossing through the boundary and the interaction between different scales,the effect of multiple scales in non-smooth systems has been one of the challenges in nonlinear dynamic fields.The main purpose of the manuscript tries to investigate the complicated dynamics as well as the evolution in non-smooth Filippov systems caused by two time scales in frequency domain,which can reveal the different modes of oscillations and non-smooth bifurcation mechanisms.Firstly,based on a Filippov system with two-dimensional predator-prey in Ecology,we introduce a periodically changed slow parameter and take suitable parameter values to establish a piecewise smooth Filippov system with two scales in frequency domain.By using the differential inclusion theory and the nonlinear dynamic theory,we analyze the non-smooth bifurcations and study the Filippov system with the non-smooth and two time scales.The results show that the system will exhibit alternate appearances of the quasi static state and large oscillations and there are several possible corresponding mechanisms under certain parameter conditions.The transition from the quasi static state to the large oscillations is caused by the delay critical bifurcation or delay Hopf bifurcation,while Hopf bifurcation induces the system to finish large oscillations and return to the quasi static state.In particular,due to the existence of non-smooth properties,the system may show some special bifurcation modes around the boundary,such as sliding bifurcation and grazing bifurcation.Secondly,By taking a boost converter controlled Filippov with three-dimensional piecewise smooth model as an example,introducing a periodically changed electric power source and taking suitable parameter values,a piecewise smooth Filippov system with two scales in frequency domain is established.The whole exciting term can be regarded as a slow-varying parameter for the exciting frequency is far less than the natural frequency,which leads to the generalizedautonomous system.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 transformed phase portrait,the associated bifurcation mechanisms of the oscillations are 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 quasi static state and large oscillations,but also influence the bifurcations between the two states.At the same time,the transformed phase portrait is an effective method about analyzing the oscillations and mechanisms of periodic excitation system with different scale coupling in frequency domain.