Dynamic Analysis Of Several Classes Of Delayed Neural Network Models

In this thesis, we deal with some important properties of the dynamic behaviors of several classes of delayed neural networks, which include the existence, uniqueness and exponential stability of the periodic solutions and the equilibrium point. The thesis is divided into six chapters.In Chapter One, the background and the history of neural networks are briefly addressed while the motivations and outline of this work are given. And some notations, definitions and lemmas are also listed in this chapter.In Chapter Two, we study Cohen-Grossberg neural networks (CGNN) with time-varying delay. Based on Halanay inequality, a continuation theorem of the coincidence degree and analysis technique, we obtain some sufficient conditions ensuring the existence, uniqueness, and global exponential stability of periodic solutions. Our results extend and improve some earlier publications.In Chapter Three, we consider the convergence dynamics of Cohen-Grossberg neural networks (CGNNs) with continuously distributed delays. Without assuming the differentiability and monotonicity of activation functions, the differentiability of amplification functions and the symmetry of synaptic interconnection weights, we construct suitable Lyapunov functionals and employ inequality technique to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, global exponential convergence, and even global exponential stability of equilibria. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria and can be applied to neural networks including Hopfield neural networks, bidirectional association memory neural networks and cellular neural networks.In Chapter Four, we study the dynamic behavior of bidirectional associative memory (BAM) neural networks with distributed delays. Based on the continuation theorem of coincidence degree theory and Krasnosel'skii's fixed point theorem on cones, we obtain some new sufficient conditions ensuring existence as well as the global exponential stability of the periodic solution. Our results are milder and less restrictive than previous known criteria since we drop the hypothesis of boundedness and differentiability on the activation function. In addition, these theoretical results are verified by numerical simulations.In Chapter Five, we consider the dynamic behavior of Cohen-Grossberg neural networks with delay. We consider non-decreasing activations which may also have jump discontinuities in order to model the ideal situation where the gain of the neuron amplifiers is very high and tends to infinity. Some sufficient conditions are given to guarantee the existence, uniqueness, and global stability of the equilibrium point. Convergence behaviors for both state and output are discussed. The constraints imposed on the interconnection matrices concern the theory of M—matrices, and are independent of the delay parameter and easily verifiable. Some existing results are improved and extended. The theoretical analysis are verified by numerical simulations.In Chapter Six, we study the existence and global exponential stability of periodic solutions for a nonlinear periodic system, arising from the description of the states of neurons in delayed Cohen-Grossberg type. We consider non-decreasing activations which may also have jump discontinuities in order to model the ideal situation where the gain of the neuron amplifiers is very high and tends to infinity. Under suitable assumptions on the interconnection matrices, we deduce some sufficient conditions ensuring existence as well as global exponential stability of periodic solution. The presented conditions concern the theory of M-matrices and are easy to check. Furthermore, due to the possible discontinuities of the activations functions, we introduce a suitable notation of limit to study the convergence of the output of the delayed neural networks. An numerical example is given to illustrate the theoretical results.