Dynamics Of Four-dimensional Neural Networks With Time Delays

Dynamics of neural networks has been center of mathematics, mechanics, information science, physiological science, computer science, cognitive science, and so on. With the extensively and important development of delayed neural networks in the fields of pattern recognition, optimizations, signal processing, control, artificial intelligence, and etc, the issues of the dynamical characteristics of delayed neural networks have attracted worldwide attention in recent years. As we know, delayed neural networks can exhibit various dynamical behaviors, such as stable, unstable, oscillations, and chaos. The investigation of these rich dynamical characteristics provides theoretical foundation and technological ways in the wide applications of delayed neural networks. However, the theoretical analysis of dynamics of delayed neural networks is a hard problem because the solution space of system is of infinite dimensions. Hence, the study of dynamics of delayed neural networks is very challenging and involves important scientific significance and application values.This dissertation focuses on the stability and bifurcation of four-dimensional neural networks with time delays. The main themes and contributions of the dissertation include:1. A delayed Hopfield neural network of four neurons with a short-cut connection is constructed by considering the small world effect of complex networks. This network with multiple time delays is reduced to the model with single time delay under the simplified assumption. The local stability and Hopf bifurcation of the model are analyzed by discussing the distributions of roots of the associated characteristic equation. The propertiees of bifurcated periodic solutions are determined by means of the center manifold theory and the normal form approach. The sufficient conditions for the global continuation of periodic solutions arising from local Hopf bifurcation are obtained according to the global Hopf bifurcation theorem. Numerical simulations are performed to validate the theoretical analysis. Effects of the short-cut connection on the dynamical behaviors of the network are discussed.2. The delayed neural network without any assumption on the short-cut connection is simplified to the model with two time delays by proper variable transformation. By analyzing the associated characteristic equation with two time delays, the sufficient conditions for the local stability and Hopf bifurcation of the network are obtained. The formulae for the propertiees of bifurcated periodic solutions are given according to the center manifold construction and the normal form computation. The existence of non-trivial equilibrium points and pitchfork bifurcation are shown. Numerical examples are given to validate the theoretical analysis. Some discussions of the effects of the short-cut connection on the dynamical characteristics of the network are shown.3. A bidirectional delayed neural network consisting of four neurons is constructed. The criterion of the global asymptotic stability of the network is derived by constructing a suitable Lyapunov functional. The distributions of roots of the associated characteristic equation are discussed by means of space decomposition, and then the sufficient conditions of the local stability of the network are given. Nontrivial synchronous/asynchronous equilibrium points and periodic solutions arising from standard/equivariant pitchfork and Hopf bifurcation are shown, respectively. The existence of co-dimension two bifurcation points is discussed. Numerical simulations are performed to show the nontrivial synchronous/asynchronous equilibrium points, periodic solutions, and multi-stability.4. A circuit experiment is constructed by time delay circuit and neural network circuit in order to support the pervious analysis. Time delay circuit is achieved by using an analog-to-digital converter, a digital signal processor and a digital-to-analog converter. It can generate different and adjustable time delays. The neural network circuit is constructed based on the Hopfield neuron and nonlinear neuron transfer function circuit. Moreover. Experimental results reach an agreement with previous conclusions.