Research On Nonlinear Dynamic Behavior Of The Parametrically Driven Stochastic Duffing-Van Der Pol System

The research in this paper is based on the famous Duffing-Van der Pol(DVP) system which has broad practical background. The complex non-linear dynamics characteristics of parametrically driven DVP system are discussed through numerical method. Further, the research focuses on the complex non-linear dynamics behaviors of parametrically DVP system with random parameter and the influence of stochastic intensity on them. The main contents of the paper are as following:Firstly the symmetry breaking bifurcation of chaotic attractors of parametrically DVP system is explored based numerical method. According to a variety of non-linear behaviors, such as symmetry breaking bifurcation, coexistence of periodic and chaotic attractors, chaotic internal crisis and boundary crisis and merging crisis, the symmetry invariance properties of the non-linear symmetry system are fully been show. The dynamics responses of non-linear symmetry system always keep symmetry, whatever they are periodic phase trajectories, chaotic phase trajectories, attractors or attraction basins. However there are various symmetry forms depending on different conditions, the transition of different symmetry forms will lead to symmetry breaking phenomena. The results in the paper extend the knowledge of symmetry breaking phenomena which were limited to periodic phase trajectories.Secondly according to orthogonal polynomial approximation theory, theoretical derivation on parametrically driven DVP system is made by adopting the second kind of Chebyshev polynomial, and the corresponding equivalent deterministic extended-order equation set is obtained. Based on the equivalent deterministic extended-order equation set, the complex non-linear dynamical behaviors of stochastic DVP system are discussed such as bifurcation, crisis and coexistence of different attractors. It is seen from the results that similar to deterministic system, the parametrically driven stochastic system has complex non-linear dynamical behaviors, and the responses of symmetry stochastic system have various symmetry forms, in the transition of which plenty of symmetry breaking phenomena occur.In addition, the changes of the responses of the parametrically DVP system with random parameter under different stochastic intensity are emphatically studied. The varying of stochastic intensity result in the visible change of the system’s dynamical characteristics, which are obviously distinguishing compared with deterministic system. Stochastic intensity being a bifurcation parameter, the responses of the system occur various bifurcation and symmetry breaking crisis phenomena.The whole research in this paper is summarized in the end, and some expectations about further research are put forward.