Bifurcation And Control Of Nonlinear Dynamical Systems

As a leading research field in nonlinear science, bifurcation and control has become more and more challenging. Bifurcation control aims at designing a controller to modify the bifurcation properties of a given nonlinear system, which could achieve some desirable behaviors. In this thesis, based on a complete summary of the history and actuality of the bifurcation and control, the author investigates further bifurcation and control of nonlinear dynamical systems by employing the nonlinear dynamics theory, the nonlinear control theory and the bifurcation theory. The organization of this paper is as follows:In the first Chapter of this dissertation, the current status about bifurcation and control of nonlinear dynamical systems, especially delayed dynamical systems. Furthermore, the author introduces the main contents and originalities of this paper.The second Chapter investigates the multi-parameter Hopf bifurcation for a class of Hopfield neural networks with three delays. Moreover, the author chooses the combination of multiple coefficients as bifurcation parameter, and derives the sufficient conditions of local stability, pitchfork bifurcation and Hopf bifurcation.The third Chapter studies bifurcation and control of a class of R?ssler chaotic system. The author proposes a double feedback controller, which can not only advance or delay bifurcation effectively, but also control chaos.The fourth Chapter discusses local stability and Hopf bifurcation of Van der Pol-Duffing equation with delayed feedback control. Also, bifurcation direction and stability of periodic solutions are analyzed.The fifth Chapter considers Hopf bifurcation control of one dimension small-world networks, and proposed parametric delay feedback controller. Choosing delay as bifurcation parameter, the author studies Hopf bifurcation of controlled small-world networks with delay dependent coefficients. By introducing some proper slowly varying parts into the bifurcation parameters, the stability can be improved.The sixth Chapter summarizes the research work of this dissertation. Furthermore, the future research direction is made.