The Dynamic Behavior, Chaos Control And Synchronization In The Duffing-Van Der Pol Oscillator Under Parametric Excitations

It is known that the famous Duffing-Van der Pol system has broad practical background. In this paper, the problem of chaos control and chaos synchronization of the Duffing-Van Der Pol oscillator under parametric excitations are investigated. In the course of their parametric excitation increasing, using a variety of numerical analysis methods found many complex dynamic behavior of the systems. This includes quasi-periodic, period, period-doubling bifurcation the formation of chaos, chaotic behavior and periodic windows alternately, as well as periodic and chaotic attractors co-exist. Found a more extensive symmetry breaking crisis. In the chaotic and periodic attractors coexistence region, the formation and crisis of the chaos attractor were studied by the changes of the coexisted attractors and corresponsing attraction domain. Further study of two coupling parametrically excited Duffing-Van der Pol system. Selecting the initial conditions in the original period, chaotic or coexistence region, by changing the initial conditions and coupling coefficient, to achieve synchronization and draw the phase diagram and the synchronization diagram. Finally, we bring out a new understanding of chaos synchronization.In this paper, the Duffing-Van der Pol system under parametric excitation dynamics and chaos synchronization was studied. Its research is not only important theoretical significance, but also has some prospect, Paper is divided into six chapters, as follows:The first chapter introduces the development background of chaos theory, research contents, methods, and this topic Duffing-Van der Pol system description, significance and innovation and so on.The second chapter the part of preliminary knowledge briefly introduces some definition of chaos, of the commonly used methods, basic knowledge, chaos control and synchronization of the background and meaning.The third chapter analyzes complex chaotic system dynamics, within a certain range of parameters by changing the size of the system equations, numerical methods, the phase diagram for the system, Poincare sections, fast Fourier transform FFT graph, bifurcation diagrams and the maximum Lyapunov index map, etc., we knew in which parameters, the system is chaotic or period initially. And found in the bifurcation process of symmetry breaking phenomenon.The fourth chapter study the coexistence of chaotic and periodic phenomena of the parametrically excited Duffing-Van der Pol system near the specific parameters. By the changes of the coexisted attractors and corresponsing attraction domain in the system, the formation and crisis of the chaos attractor were studied. The phenomenon of symmetry breaking has also been a further understanding.The fifth chapter focusing on two coupled parametrically excited Duffing-Van der Pol system, analyzed and summarized the problem of chaos Synchronization.The last chapter summarizes the work on the text, and made some problems for further study.