Dynamical Behaviors Analysis Of Filippov System

Since there exist many non-smooth factors in science and engineering such as shock,dry friction and switches,the corresponding dynamical behavior can be described via establishing models with non-smooth terms.Since there may exist much complex dynamical behavior in non-smooth systems,those methods for the smooth systems cannot be applied and new theory should be developed to reveal the mechanism.Most of the recent works focused on the non-smooth system are with single scale.Dynamical behaviors of the systems with multiple scales,however,may appear to be complicated,for which the non-smooth dynamics with multiple scales coupling have attracted much attention.In order to explore the dynamics with two scales of the non-smooth Filippov systems,the bursting oscillations as well as its mechanism are investigated in this paper,and the evolution of the dynamics is explored by employing the corresponding theory.Firstly,a typical type of Filippov system with the periodic external excitation based on a class of three-dimensional chaotic systems,namely Liu and Glukhovsky-Dolzhansky system,are investigated.By introducing periodic excitation and non-smooth terms,the new Filippov-type models with two scales are established.Via numerical simulations,the periodically changed external excitation in the modified Liu system may cause three different sliding movements when crossing the boundary.The evolution from the non-smooth quasi-periodic oscillations to different types of bursting attractors can be observed in the Glukhovsky-Dolzhansky system.Via employing the differential inclusion theory,the conditions of smooth and nonsmooth bifurcations are derived.With the parameters fixed,the bursting oscillations can be observed through numerical simulation.By introducing the superimposing between the transformed phase portrait and the equilibrium branches,the mechanism of different bursting oscillations can be revealed.Secondly,a Filippov system with the periodic external excitation as well as the parametric excitation based on a class of three-dimensional chaotic systems is studied.On the basis of the Liu model,parametric excitation is introduced to construct a new Filippov system.We analyze the bifurcation that may occur in the modified system and the condition of those bifurcations.Via changing the amplitude value A and fixing the remaining parameters,the equilibrium branches can be obtained.Hence,the mechanism of different types of bursting attractors can be revealed via superimposing the transformed phase portrait and the equilibrium branches.In particular,the period doubling bifurcation of limit cycle will lead to the change of structure of the bursting attractor,where the period of oscillations is almost doubled in the spiking state.