Hopf Bifurcation Control And Period-doubling Bifurcation Anti-control Of Several Dynamical Systems

Bifurcation control of nonlinear dynamics has become a hot research topic,and the research on bifurcation control of high dimensional systems is very difficult.There are a lot of bifurcation phenomena in nature and engineering.In order to avoid the harmful situation,or the need to strengthen the beneficial bifurcation and to occur bifurcation,bifurcation characteristics of the system need to eliminate the delay,and other ways to control,which is the main research purpose and content of the theory of bifurcation control.In this paper,mainly study and analyze the Hopf bifurcation control problem of high dimensional chaotic nonlinear system,and the problem of the period doubling bifurcation anti-control of Logistic system.In this paper,the research contents,current situation and development trend of nonlinear control theory,bifurcation control,Hopf control and bifurcation control of discrete systems are summarized.Then,the basic concepts,the definition and classification of nonlinear dynamical systems are introduced.In this paper,the definition and criterion of Hopf bifurcation,Hopf bifurcation theory,the method of solving Hopf bifurcation periodic solution,the limit cycle stability index of Hopf bifurcation and the approximate analytical solution of amplitude are given.This paper focuses on the Hopf bifurcation analysis and control of high dimensional hyperchaotic nonlinear systems.For a class of Lorenz-Like three dimensional hyperchaotic system,by using the normal form and high dimensional bifurcation theory,analyzes the existence and stability of the Hopf bifurcation;A new dynamic state-feed-back control law is proposed,and the Hopf bifurcation control of the system is carried out by the new method.By changing the control parameters,the Hopf bifurcation of the system can be advanced or delayed,and the bifurcation characteristics of the system can be changed;The formula for calculating the amplitude of four dimensional nonlinear dynamic system is presented.The amplitude control of the system is carried out,and the amplitude control relation of the system is obtained.The Hopf bifurcation and control of a four dimensional autonomous uL && system are studied.The stability of the unique zero equilibrium point of the system is judged by the Routh-Hurwitz theory;The existence of the system Hopf bifurcation is analyzed.The critical values of the Hopf bifurcation parameters are obtained.The stability of periodic solutions of Hopf bifurcation is analyzed by the Hopf bifurcation theory;The state feedback control method is used to control the Hopf bifurcation of the system,setting up the combination controller composed of linear and nonlinear controller.By controlling the control parameters,the Hopf bifurcation control and bifurcation periodic solution stability control are realized.The correctness of theoretical analysis has been verified by numerical simulation.The problem of stability and Hopf bifurcation of a five dimensional hyperchaotic system solutions are analyzed,obtained the first Lyapunov coefficient,given Hopf bifurcation stability conditions;the stability of control system is a nonlinear controller.The relationship between the control parameters and the first Lyapunov coefficients is obtained by theoretical derivation,and the change of the control parameters can be controlled to realize the change of the stability range of the Hopf bifurcation.The correctness of the theory is verified by numerical simulation,and the range of control parameters is obtained.The existence of Hopf bifurcation for a Pan-Like hyperchaotic systems is carried out,by the high dimensional Hopf bifurcation theory and Routh Hurwitz theory.The Hopf bifurcation characteristics of the system are analyzed Hopf bifurcation control system using hybrid control method,set up the nonlinear controller changes the system Hopf bifurcation,and to control the stability of the system are obtained the relationship between the stability of the system parameters and control parameters.Numerical simulation is used to prove the correctness of the theoretical analysis.It is also an important part of the research on bifurcation control of discrete systems.Set of linear and nonlinear controller for discrete systems Logistic,two period and four period doubling bifurcation anti-control,the corresponding relationship between the control parameters and the bifurcation parameter,by controlling the parameters change,produce can achieve the desired bifurcation.The main content of this paper is to study the Hopf bifurcation control of high dimensional nonlinear dynamical systems and the period doubling bifurcation of discrete system,which enriches the research contents of Hopf bifurcation control,and has great guiding significance in theory and practice.