Feedback Control Of Hopf Bifurcation In Three Dimensional Nonlinear System

As a new frontier topic,the bifurcation control of 3D and high-dimensional systems is extremely challenging.Bifurcation control of low-dimensional systems has been studied to some extent,but the difficulty of bifurcation control in 3D and high-dimensional systems is much more difficult than the bifurcation control of low-dimensional systems.Eliminating and delaying the occurrence of bifurcations so as to avoid bad consequences and to form and strengthen beneficial bifurcations so that they can be utilized by people is a major element of bifurcation control.In this paper,we mainly study the Hopf bifurcation limit-cycle control of a typical three-dimensional nonlinear system and a nonlinear system whose limit cycle has three dimensional characteristics.These studies include analyzing the bifurcation characteristics,offering approaches of control,designing bifurcation controllers and thereby achieving some desirable dynamical behaviors.This paper first reviews the research status of the analysis of system control theory,bifurcation control theory,Hopf bifurcation control,chaos control,high-dimensional nonlinear control and so on.Then some basic theories of bifurcation control theory are introduced.The stability theory,the concept of bifurcation,the content and method of bifurcation theory,Hopf bifurcation theory and Hopf bifurcation control method are introduced.Then the article gives the basic theory of limit cycle amplitude control of three-dimensional nonlinear system,including approximate solution of limit cycle amplitude,criticality of Hopf bifurcation,design of controller,the relationship between the controller and the limit ring amplitude.As an important research direction of Hopf bifurcation control,this paper mainly studies the limit-cycle amplitude control of three-dimensional nonlinear systems.Based on the central popular theorem and the normative dimensionality-reduction theory,aneffective control limit cycle method is proposed.For general three-dimensional systems,this paper studies the design method of universal polynomial controllers for standard subsystems.In this paper,a three-dimensional Langford system is selected from the typical three-dimensional nonlinear system to study the bifurcation control.In this paper,the bifurcation analysis of Langford system is firstly carried out and then the controller is designed according to the general theory of controller design.Then the MATLAB simulation proves that the controller has an effective control effect on the system limit cycle amplitude.This paper also studies the bifurcation control of a new three-dimensional nonlinear system.One of the big differences between the system limit cycle and the Langford system limit cycle is that the limit cycle has three-dimensional characteristics.First of all,this paper has carried on bifurcation analysis to this system.This analysis determines the system parameters when the system limit cycle exists.Then the controller was creatively designed to realize the amplitude control of the limit cycle of the system.The research in this paper enriches the research content of nonlinear bifurcation control theory and contributes to the research of bifurcation control of three-dimensional nonlinear system,which has great theoretical significance and practical application value.