Control And Anticontrol Of Chaos Based On State Feedback And Adaptive Method

Since the first chaotic attractor was found, chaos has been attracting more and more attention of researchers in various fields of science and becoming a challenging and promising research field. In the past three decades, the research of chaos gain grounds, and many different chaos control methods have been proposed. Chaos control can be classified into two categories:one is to suppress the chaotic dynamical behavior if it is harmful, and the other is to generate or enhance chaos if it is useful, namely, chaos anticontrol. The main results contain as follows.The research history and development of chaos are briefly retrospected. The definition and properties of chaos are introduced. Meanwhile, the basic mathematic theory and method for chaos research are proposed. The basic objective of chaos control and the existing control methods are introduced. A few stability theory and dynamical systems theory are proposed for the further research in this paper.Based on the Routh-Hurwitz criterion, the stability of a new chaotic system is investigated. The spectrums of maximal Lyapunov exponent are presented for the dynamic analysis. State feedback controllers are designed to control the chaotic system to the equilibrium points and limit cycle. Theoretical analysis gives the range of value of control parameter which stabilize the equilibrium points and its critical parameter for generating Hopf bifurcation. The periodic orbits are stabilized by parameter adjustment. Numerical simulations indicate that the method can effectively guide the system trajectories to the equilibrium points and periodic orbits.The dynamic behavior of the second class of nontrivial equilibrium point on4D Qi system is analyzed. The spectrum of maximal lyapunov exponent and the bifurcation diagram corresponding to the second class of equilibrium points are drawn. A parametric adaptive controller is presented utilizing the observation of the system variables. The controller can correct the perturbations of the parameter and ensure the stability of the controlled system. The Lyapunov function is used to prove that the parameter control system is asymptotically stable. The numerical simulations results indicate that the control method is of very quick convergent speed.The Hopf bifurcation control of a new hyperchaotic Lu system is studied. The stability of the hyperchaotic Lii system depending on a selected control parameter is studied, and the critical value of the system parameter at which Hopf bifurcation occurs is investigated. Theoretical analysis gives the stability of the Hopf bifurcation. In particular, Washout filter aided feedback controllers are designed for delaying the bifurcation point and ensuring the stability of the bifurcated limit cycles. An important feature of the control laws is that they do not result in any change in the set of equilibrium. Computer simulations are presented to confirm the analytical predictions.The anticontrol of Lorenz system is studied. A new chaotic system which enhances the chaotic behavior of Lorenz system is constructed by using a state feedback controller. The local bifurcation of the non-hyperbolic equilibrium point of the new system is investigated by using center manifold. The bifurcation diagram and spectrum of maximal Lyapunov exponent are presented for the dynamic analysis. Moreover, an electronic circuit is designed for verifying the chaotic behavior of the new system. The circuit simulations are found to be identical to the numerical simulations, and the new chaotic system can be realized.