Non-smooth Bifurcation Analysis Of A Class Of Filippov System With Two Scales

Because of the jumping phenomena of the vector field on the non-smooth boundary,some special oscillating behaviors can be observed in Filippov dynamical system,such as sliding,grazing and so on.Meanwhile,the coupling between multiple scales in frequency domain may lead to different forms of bursting oscillations.The effect of multiple scales in Filippov dynamical system becomes one of the hot and frontier subjects.In this paper,the thesis mainly take the coupled BVP circuit system as an example and a Filippov system with two scales in frequency domain is established by introducing suitable parameter values.The typical complicated bursting oscillations as well as the mechanism in different equilibrium states are revealed by utilizing the theory of nonlinear dynamics bifurcation,the fast-slow analysis,numerical simulation,etc.Firstly,based on the coupled BVP circuit system which contained a single non-smooth boundary,by selecting appropriate parameters to make the exciting frequency far less than the natural frequency,a Filippov system with two frequency scales is established.We consider the bursting oscillations and bifurcation mechanism of a class of Filippov system in a single equilibrium state.The stability of equilibrium and conventional bifurcations for two smooth subsystems are respectively analyzed with the theory of nonlinear dynamics.Using the method of slow-fast analysis,bifurcation diagram of equilibrium has been overlapped with phase diagram to discuss the mechanism of different bursting oscillations.Meanwhile,the potential unconventional bifurcations and their existing conditions on the non-smooth boundary are studied by the theory of differential inclusion.It is found that with the change of parameters,the transition between quiescent state and spiking state is mainly caused by non-smooth factors in different bursting phenomenon.For three typical cases of external excitation amplitude,typical periodic oscillations with sliding structure are given.The dynamical mechanism of the transition between quiescent state and spiking state is revealed when the system trajectory is not touching,touching but not crossing,touching and sliding then crossing the non-smooth boundary.Secondly,suitable parameters are introduced to obtain multiple equilibrium states for the same circuit model as previous chapter.The bursting oscillations and the mechanism of one class of Filippov system with multiple equilibrium states arefurther researched combining multiple scale factors.Regarding the whole exciting term as a slow-varying parameter,we obtain the equilibrium branches as well as the bifurcation details and investigate the influence of exciting amplitude to oscillating behaviors.Two typical cases are focused in the paper to present different forms of bursting oscillation.With the method of slow-fast analysis and dynamic characteristics of the vector field on the non-smooth boundary,based on the transformed phase portraits,the mechanism of oscillations is presented respectively.It is pointed out that more complex bursting oscillations may be exhibited in multiple equilibrium states.In addition,at some special points on the equilibrium branches,spiking oscillations can be found.With the increase of the amplitude of the external excitation,when the related equilibrium branches pass across the special points,bursting oscillations can be observed.Different from smooth dynamical systems,the spiking oscillations in Filippov system behave in alternations between the sliding movement along the non-smooth boundary and the relatively large amplitude oscillations,the mechanism of which is presented according to the characteristics of the equilibrium branches as well as the change of the vector field on both sides of the non-smooth boundary.Finally,the main work of this paper are summarized.Some related shortages are pointed out and the future research direction is proposed.