Investigation Of Two Types Bifurcation Control And Chaos Control For Nonlinear Dynamical Systems

As an engineering leading research field, bifurcation control has become more and more challenging. It aims at designing a controller to modify the bifurcation properties of a given nonlinear system, and achieving some desirable behaviors. Through a complete summary and examination of the history and the actuality of the bifurcation control research, a systematic investigation into the fundamental theory and application of the bifurcation control has been conducted based on the nonlinear vibration theory, the nonlinear dynamics theory. The studies have more profound theoretical significances and important engineering application values, which contribute to the development and application of period-doubling bifurcation control,Hopf bifurcation et al. The main contents in this paper can be stated as follows:Chapter one outlines some study methods and recent advances about bifurcation control and nonlinear theory first, and then introduces the study's aim, contents and innovations.Chapter two introduces some basic conceptions about dynamics. The occurring conditions of three elementary static bifurcations, including saddle-node bifurcation, transcrytical bifurcation and pitchfork bifurcation, and transfer condition of each other are introduced. Then some design and analysis methods about bifurcation controller are presented.In chapter three, linear controller and nonlinear controller are designed to period-doubling bifurcation of logistic model. The bifurcation maps of logistic model are achieved and the bifurcation characteristic of the dynamical system is changed by using variables parameterized controller. In order to change the parameter value of all existing bifurcation point and modify the shape of type of a bifurcation chain, we can design variables controller in view of practical aims. Moreover, the bifurcation map may get more effective by using controller in an advisable way.In chapter four, state feedback controller and washout filter controller are applied in controlling the amplitude of Hopf bifurcation of van der Pol-Duffing system. The effects of controlling parameters to the amplitudes of limit cycles are gained by analyzing the controlling equations, so the control strategy is advanced. The desirable state feedback controller is designed and the Hopf bifurcation is controlled well. In chapter five, linear state feedback controllers are designed to control the equilibrium points and period trajectories of Lorenz system. Firstly, we apply Routh-Hurwitz criterion analyzing the stability of the controlled system. The choice principle of feedback coefficients to attain control objective is proved strictly, then numerical simulation results is indicated that the method can effectively guide the system trajectories to equilibrium, specially, the method can direct the controlled system to 1 periodic trajectories also, we obtain the reigion of controlling parameter of 1 periodic trajectories.In this paper, the innovative is that the bifurcation control theory is used to investigate the nonlinear dynamical systems, which enriches the nonlinear dynamics theory and expands the nonlinear control theory. The creative things are as follows: apply the feedback control technology to the period-doubling bifurcation in logistic model; control the Hopf bifurcation of van der Pol-Duffing system by washout filter controller for the first time; mobilize linear feedback control to control the chaos of equilibrium points and 1 period trajectories of Lorenz system.