The Stability Analysis And Bifurcation Control Of Nonlinear Dynamic System

Bifurcation and control is a leading research field in nonlinear science, there are manycomplex dynamic behavior exist in dynamical system. In the actual project, some of thesedynamics behavior are beneficial but some others are harmful. Bifurcation control aims atdesigning a controller to modify the bifurcation properties of a given nonlinear system, whichcould achieve some desirable behaviors. In this thesis, introduced the history and actuality ofthe bifurcation and control, based on this knowledge, we studied the codimension-twobifurcation, hyperchaos, and bifurcation of the hybrid control.The author investigates further bifurcation and control of nonlinear dynamical systemsby employing the nonlinear dynamics theory, the nonlinear control theory and the bifurcationtheory. The organization of this paper is as follows:In the first Chapter of this dissertation, state the current status about bifurcation andbifurcation control of nonlinear dynamical systems, especially made the brief summary anddescription to the control of Hopf bifurcation. Furthermore, we introduce the main contentsand originalities of this paper.The second chapter introduces the basic Hopf bifurcation theory of the dynamic system，and introduces a method to study codimension-two degenerate Hopf bifurcation, we calledLyapunov constant method, and then introduces some bifurcation and chaos theory we needin this paper. Including the definition of bifurcation and chaos, the main characteristics andthe main methods we need in this article. Finally introduces a main method of bifurcationcontrol, called hybrid control method (the parameter perturbation control combined withnonlinear feedback control). All these theories laid the foundation for later study, so that wecan better to analyze the bifurcation control of nonlinear dynamical systems.The third chapter proposed a new Liu chaotic system. Parameter conditions forsupercritical, subcritical and the codimension-2degenerate Hopf bifurcation are presented byanalyzing the first Lyapunov coefficient of the chaotic system. Then a control items add to thenew chaotic system, we obtain a four dimensional hyperchaotic system. Application ofnumerical simulation verified the hyperchaos dynamics characteristic of the system, there aretwo positive Lyapunov exponents. The stability and direction of limit cycle are derived bycenter manifold theorem and normal form theory. The correctness of theoretical derivation areverified by numerical simulation.In the forth chapter, on the basis of hybrid control strategy we introduced in chapter2,we study the bifurcation behavior of Chua’s circuit system which containing a nonlinearfunction. Proposing the dynamic model, a hybrid control item is added to the first branch andthe second branch of the system in turn. Through the parameter perturbation and the nonlinear feedback control study the bifurcation behavior of the system. Through theoretical analysisand numerical simulation the expected bifurcation characteristics obtained by the hybridcontrol, we find that this control strategy is very effective.