Bifurcations And Chaotic Theory Of Nonlinear Dynamical Systems And Its Application

In this thesis, by the theory of bifurcations and chaos, we at first investigate bifurcations and chaos of a delayed ecological model. Then, we discuss the chaos control and synchronization in a modified Chen system and a modified Lü system, respectively. Also, we study the application of chaos control and synchronization in secure communication.The layout of this thesis is as follows.In Chapter 1, a brief review concerning the theory of local bifurcation and chaos of nonlinear dynamical systems is introduced.In Chapter 2, a complicated delayed ecological model is studied in detail. First, the parameter plane is divided into three different regions, in which there exist various fixed points of the model. Second, by using the center manifold theorem, it has been proven that there exists the transcritical bifurcation rather than the saddle-node bifurcation with respect to the zero fixed point. For the nonzero fixed points of the model, their stabilities and bifurcations are studied when some parameters are varied. Third, the numerical simulation results not only show the consistence with the theoretical analysis but also display some new interesting dynamical behaviors. In particular, some incorrect observations, such as a 3-cycle and many 4-cycles, obtained only from the bifurcation diagram are revised by our analysis. Finally, the computations of the maximum Lyapunov exponents confirm the chaotic behaviors.In Chapter 3, an approach, based on the observer design, is introduced to achieve the synchronization of two modified Chen systems. The two synchronized systems are used as the transmitter and the receiver of a secure communication system, respectively. Furthermore, two methods are used to inject the information signal into the chaotic drive-system at the transmitter. In particular, we obtained, in whichever method the information signal is added, the synchronization of the transmitter and the receiver is still maintained, which also further verifies that the synchronization based on the observer approach is a robust phenomenon. In addition, two examples of signal transmission based on a pair of synchronizing modified Chen's systems are applied to present the effectiveness of the method in secure communication.In Chapter 4, by replacing a quadratic nonlinear term in Lü system with a piecewise linear signum function, a new simplified three-dimensional piecewise con-...