Feedback Control Of Hopf Bifurcation In Two Classes Of Nonlinear High Dimentional Systems

As an emerging leading research field, bifurcation control of high dimensional systems has become more and more challenging. Some fruits have been achieved in low dimensional systems and bifurcation control of high dimensional systems is quite difficult than that in low ones. It is the main topics in bifurcation control to either delay (or eliminate) bifurcations for avoiding bad affects or creat (or enhance) beneficial bifurcation behaviors on purpose of utilizing them. This paper focuss on studying the Hopf bifurcation control of the typical 3D and 4D nonlinear systems, e.g. analyzing the bifurcation characteristics, offering approaches of control, designing bifurcation contollers and thereby achieving some desirable dynamical behaviors.Firstly, the recent advances about the nonlinear control theory, the bifurcation control, the Hopf bifurcation control and the chaos control are summarized in this paper. Secondly, some basic concepts about nonlinear dynamics and several bifurcation control methods are introduced, the definition, the criterion and the theory of Hopf bifurcation as well as the approach of the periodic approximate solutions are illustrated and simutanously the formulae for stability index and analytical amplitude approximations of limit cycle are presented.As an important aspect in Hopf bifurcation control, the amplitude control of limit cycle is stressly investigated in 3D and 4D nonlinear systems. A convenient approach to achieve an effective amplitude control is proposed from the center manifold theory and normal form reduction.For a 3D system, formulae for a general polynomial controller applied to the normal form of a system and for the relvant stability index are derived. Thereby the amplitude approximation in terms of control gains of the controlled system is solved. The Chen system and smooth Chua's equation are used to illustrate the application of the amplitude control technology given above and show that the control technology is convenient and valid. The Langford system, which is the same as the Chen and Chua's system in being 3D system but different from them in amplitude control application, can be controlled in amplitude of limit cycle directly and effectively and the control gains can be optimized conveniently. The controller employed in the original Langford system can be directly designed for a desirable amplitude target and the exact solutions for amplitudes of limit cycles in systems under some special controllers can be obtained. For modifying amplitude of limit cycle in the 4D Qi system, the washout filter controller is applied in the feedback nonlinear control and adds the dimension of Qi system to SD, the analytical amplitude approximation in terms of control gains is well established and the curves showing that result of amplitude control are given with different choices of control gains. The analytical amplitude approximations are all compared with the numerical simulations that it is varified that the errors are small enough in vicinity of Hopf bifurcation.The first and secondary Hopf bifurcation and the chaos motion of the Langford system are controlled and the relationships between bifurcation parameter and control gains are obtained. The first Hopf bifurcation points are shifted under a state feedback linear control while the secondary Hopf bifurcation value is changed preserving one of the first Hopf bifurcation points under a nonlinear control. The chaos motion is delayed by the way of controlling the quasi-periodic bifurcation route to chaos.Anti-control of Hopf bifurcation of the 4D Qi system is investigated. The Hopf bifurcation behavior from the zero equilibrium, which is a saddle point, is created under the combination control of linear and nonlinear feedback. With respect to various bifurcation parameters, different approaches of anti-control are advanced.The research work enriches the nonlinear control theory, develops the technique of bifurcation control, contributes to the establishment of bifurcation control theory of high dimensional systems and has great theoretical meanings and pratical values.