Chaotic Synchronization And Bifurcation Control For Several Classes Of Nonlinear Systems

Chaos and bifurcation are important theories in the nonlinear field.Under certain conditions,the phenomenons of chaos and bifurcation are bound to occur in the nonlinear system,and the stability of the system will be affected by them.Therefore,it is particularly important to study the control of chaos and bifurcation in the nonlinear system.With the rapid development of modern science and technology and the wide application of chaotic synchronization,chaos synchronization and bifurcation control in the nonlinear system have attracted wide attention in the academic community.In this paper,chaotic synchronization and bifurcation control for several classes of nonlinear systems are studied,including synchronization of chaotic systems and static bifurcation and control in the nonlinear systems.The main contents are summarized as follows:1.Exponential H_? synchronization for discrete chaotic neural networks with time delays and stochastic perturbationsThe problem of exponential H_? synchronization of discrete chaotic neural networks with time delays and stochastic perturbations is studied.First,by using the Lyapunov-Krasovskii functional and output feedback controller,we establish the H_? performance of the mean square exponential synchronization of master-slave systems,which is analyzed using a matrix inequality approach.Second,the parameters of a desired output feedback controller can be achieved by solving a linear matrix inequality.Finally,one simulated example is presented to show the effectiveness of the theoretical results.2.Exponential synchronization of discrete chaotic neural networks with time delay and stochastic missing dataThe exponential synchronization problem of two different discrete chaotic neural networks with time delays and stochastic disturbances is discussed.By utilizing the Lyapunov-functional approach and the stochastic analysis theory,a sufficient condition for the error dynamic system to be mean-square exponentially stable is first obtained.Then based on such sufficient condition,a reliable controller is designed to guarantee that two different discrete-time delayed neural networks with stochastic disturbances are exponentially synchronized in the mean square.The parameters of a desired state feedback controller can be achieved by solving in terms of linear matrix inequality.Finally,a numerical example is presented to validate the feasibility and effectiveness of the proposed synchronization approaches.3.Mean-square exponential synchronization of chaotic neural networks with probabilistic delay.The exponential synchronization problem of a class of chaotic neural networks with stochastic and distributed delays is concerned.Considering the probability distribution of the time-varying delay,a random variable is established to satisfy Bernoulli random distribution,and a new system is obtained to contain probability distribution information.By utilizing appropriate Lyapunov-Krasovskii functional,the Jensen integral inequality theory and linear matrix inequalities(LMIs)technique,several delay-dependent sufficient conditions are developed to guarantee the mean-square exponential synchronization of mixed time-delay chaotic neural networks with restricted perturbations.The derived criteria are expressed in terms of linear matrix inequalities,which can be easily solved by using the available Matlab LMI toolbox.Finally,two examples are provided to illustrate the feasibility and the effectiveness of the presented synchronization scheme.4.Linear generalized synchronization of spatial chaotic systemsThe problem of generalized synchronization is investigated for spatial chaotic systems.Based on the stability of the fixed point of a plane system,we obtain the stable domain of the space plane.According to the stable domain of the space plane,the stable domain of the coupling strength for the linear generalized synchronization of the spatial chaotic systems is determined.Moreover,the relationship between the stable fixed plane and the synchronization of the spatial chaos system is also analyzed.Finally,an example is presented to validate the scheme and the analysis.5.Spatial static bifurcation of 2D discrete dynamical systems and Its controlA unified time-delayed feed-back control method is adopted in this paper to control the spatial static bifurcation of 2D discrete dynamical systems.Transferring the existing bifurcation or producing a new transcritical,pichfork or saddle-node bifurcation,this method determines and then controls the spatial static bifurcation of 2D discrete dynamical systems.