Analysis Of Local Bifurcation And Generalized Hopf Bifurcation For A Class Of Filippov-type Systems

In recent years,piecewise smooth systems are extensively applied to many aspects on physical systems and technological devices in engineering and applied science.Some physical systems may contain different modes,and the transition from one mode to another can sometimes be idealized as an instantaneous transition,since the transition time from one mode to another is usually extremely short comparing to the time scale for the development of the state in each individual mode,which is described by an ordinary differential equation.Thus the mathematical models of these physical systems can be modeled to piecewise smooth dynamical systems.Piecewise smooth models often appear in many different disciplines,such as power electronic converters[14,13,48,5],mechanical systems with clearances or elastic constraints[54,52,62,63],earthquake engineering [6,33,35,34],stick-slip models with frictions[4,30,51,38]and many other physical processes.The bifurcation theory in smooth dynamical systems has been deeply studied.In particular,one cannot always apply these results to piecewise smooth systems due to their discontinuity.Piecewise smooth systems can exhibit all kinds of bifurcations which appear in smooth systems,but some bifurcation phenomena could behave quite differently from those in smooth systems due to the influence of the discontinuity of the system. The local bifurcations such as saddle-node bifurcation.transcritical bifurcation and pitchfork bifurcation present special features in piecewise smooth systems comparing to the counterparts in smooth systems.Especially the geometrical property of the discontinuity is involved in the study of the generalized Hopf bifurcation which yields the piecewise smooth bifurcating periodic orbits.Some bifurcations related to discontinuity which can not be seen in smooth systems,have been widely studied in piecewise smooth systems. such as border-collision[15,17],corner-collision[15],grazing bifurcation[7,13,14,36,37], discontinuous bifurcation[18,42],sliding bifurcation[19,38,39],non-smooth bifurcation [42,45]and multiple crossing bifurcation[43]and so on.The bifurcations of stationary points and periodic solutions are always pop problems in dynamical systems.We are interested in the Filippov-type ordinary differential equations conposed by two sub-linear-systems.If both stationary points of the sub-systems lie on the line of discontinuity and concide with each other,the degree of structural instability of this stationary point,on the line of discontinuity is at least 3.Therefore we need at least three parameters to study the bifurcation properties of this Filippov-type equation, which becomes part of the work in this thesis.The generalized Hopf bifurcation in 2 -dimensional nonlinear piecewise smooth systems has been investigated in many papers [60,61,62,63].But there is few results about the generalized Hopf bifurcation in 3 -dimensional nonlinear piecewise systems,which becomes an important part of this thesis.In this paper,we focus ourselves on studying stationary points,periodic solutions. homoclinic and heteoclinic bifurcations in Filippov-type systems.Our main work consists of:●We study the existence of solutions as time increasing of a 3 -parameter piecewise linear Filippov-type system based upon the theory of differential inclusion.We also studied the multiple existence and uniqueness of solutions,and the solutions with sliding mode by analyzing the properties of the vector field defined on the line of discontinuity. Finaly,we investigate the bifurcation properties of stationary solutions of this 3 -parameter piecewise linear Filippov-type system.●We study the bifurcation properties of periodic solutions with and without sliding motion,and the homoclinic and heteroclinic solutions in a 3 -parameter piecewise linear Filippov-type system.We construct the Poincarémap of each sub linear system on the line of discontinuity,in terms of which,we define a return map on line of discontinuity for the whole Filippov-type system.By searching the nontrivial fixed point of the return map we obtain the existence of piecewise smooth periodic solution without sliding motion.We also prove the existence of periodic solution with sliding motion and a special type homoor heteroclinic orbit,which may reaches the stationary point within finite time.●We study the generalized Hopf bifurcation mechanism for a 1 -parameter 3 -di-minsional nonlinear Filippov-type ordinary differential equation.We first analyze the existence of periodic solutions of the piecewise linear approaching system.Then we investigate the Poincarémap of each sub-smooth-system,which leads to the return map of the whole system.We prove that the interaction of the eigen-structure of each subsystem and the discontinuity determines the generalized Hopf bifurcation phenomena.