Dynamical Behavior And Synchronization Study For A Kind Of Discrete-time Systems With Random Parameter

As a kind of important and practical mathematical model, the discrete time system hasbeen widely used in biological mathematics, weather, physics and computational mathe-matics due to their promising potential application in many field and obtains an abundancetheoretical results. For the influence of random factors, the nonlinear stochastic discrete sys-tem model can better describe the model of real life. Therefore, the studies of dynamicalbehavior and chaos synchronization about the discrete-time system with random parameterhave more profound theoretical significance and engineering application value. Based onthis, the problem of dynamic behavior and synchronization in stochastic discrete-time sys-tem is discussed in this paper. The main works are as follows:Firstly，the formation and development process of bifurcation and synchronization aboutthe discrete-time dynamic system are summarized as well as the discrete orthogonal polyno-mial theory and the stability theorem of linear discrete dynamical system are introduced.Secondly, according to the orthogonal polynomial approximation of discrete random func-tion, the stochastic discrete-time systems are transferred into its equivalent deterministic sys-tem. Through stability theory and bifurcation condition, the asymptotic stability and Hopfbifurcation about stochastic logistic model with Poisson growth coefficient are investigated.The above mathematical analysis combining with numerical simulation shows that asymp-totic stability interval and the bifurcation critical value are affected by the stochastic intensityand standard deviation. We find that the parameter interval for asymptotic stability of non-trivial equilibrium is smaller as the random intensity and bifurcation phenomena under theinfluence of random intensity happen to drift.Thirdly, the generalized synchronization of stochastic discrete chaotic system is re-searched. In this paper, we design an active controller to solve the synchronization problem.Taking the synchronization about homogeneous and heterogeneous stochastic discrete sys-tems as illustrative examples confirm the validity of proposed control method.Finally, we give the main content, innovation points and the follow-up work.