Research On Bifurcation, Chaos And Their Control In Time-delayed Neural Network System

Researching the problems of neural network system with time-delay is one of the front edge issues in nonlinear dynamics domains. A time-delayed dynamic system having complex dynamic behaviors could depict the essence of nature phenomenon more precisely, and could have The phenomena of the Hopf bifurcation, chaos in a class of time-delayed neural network system and the problems of their control techniques are investigated in this dissertation. The main contributions of the dissertation are as following:1. A neuron equation with discrete time delay is studied. The conditions of equilibrium stability and Hopf bifurcation for this model are given by analyzing correspond transcendental characteristic equation. The stability of bifurcating periodic solutions and the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Since the neuron activation function is considered being arbitrarily, it provides a novel approach to study other delayed neural system with non-increase activation function.2. An algorithm for determining the largest Lyapunov exponent is developed by combining the predicting chaotic time series and modified matrix method for calculating Lyapunov exponents. This novel algorithm can avoid to solve the optimization problem and reducing the related computations. The problems of LEs of difference differential dynamic system are discussed preliminarily, that has been settled the basic frame for further research of the LLE estimating problems in such systems.3. The existence and boundary characteristics for attractors of nonlinear dynamic systems are discussed. By applying Hopf bifurcation theorem, the parameters for unstable periodic solution of the simple neuron equation with time-delay are calculated. The results of numerical simulation by Runge-Kutta method show that chaotic phenomenon could be generated in an autonomous system consisting of a simple neuron equation with time-delay. It implies that we can find a new kind of chaotic generators in a first order nonlinear dynamic system with time-delay. The system with two neurons is also discussed.4. The Lyapunov functional method in stability theory and its application to the neural network system is introduced. By using Lyapunov functional method, we get the general synchronization conditions in analytic form for the chaotic systemconsisting of a simple neuron with time-delay. So we can avoid the calculation of the condition Lyapunov exponents. Moreover, we have demonstrated that such a system is with infinite dimension and could be synchronized under certain conditions. It is expected that we can obtain a chaotic flow with much higher cryptic strength in a delayed neural system.5. The asymptotic stability theorem and relevant corollary for linearized nonlinear dynamic system are presented and proved, upon which a novel method for analyzing local stability of a dynamic system with time-delay is suggested. For the time-delayed system consisting of one or two neurons, we design a feedback control system and a Washout filter based system respectively. By employing the proposed stability theorem, we investigated the stability of a control system and show relevant theorems for choosing parameters of the stabilized control system. A preliminary study for the anti-control problem of bifurcations is also conducted. Washout filter based model is proposed and theoretic analyses are given.6. A feedback control method is suggested for a class of first order autonomous continuous-time neural network systems. The theorem for asymptotic stability conditions in analytic form of the control system is given. In the case that the parameters of a chaotic system are unknown, we proposed an adaptive control model and asymptotic stability conditions in analytic form of such system is deduced by employing the Lyapunov functional method. The novel adaptive control technique we proposed here also paves a new way to study of other chaotic control system with unknown parameters.