Dynamics Of Some Neural Networks And The Research On Chaos Theory

In this thesis, we mainly investigate the dynamical behavior of the Cohen - Grossberg neural networks with time delays and high-order terms, the bifurcations of some associative memory, and the relations between the stochastic properties and Devaney chaos's properties. The Cohen-Grossberg neural networks were firstly proposed as a class of artificial neural networks described by the ordinary differential equations in the last century. Many models from neurobiology, biological populations, and evolution theory, such as the Hop-field neural network (HNN), the bidirectional associative memory neural network and the cellular neural network, are special cases of the Cohen-Grossberg neural network (CGNN). Both CGNN and HNN have been intensively used in real application, such as parallel computation, signal processing, associative memory, and so on, which has become an attracting and focused topic. Considering the delay property and the advantage of the models with high-order terms, we discuss the existence and stability of periodic solutions in the delayed neural networks with high-order terms.The associative memory is a kind of the neural networks with practical applications. The stability of its fixed points is the focus of many researches. This thesis reports a multivalued associative memory model admitting the architecture of fully connected Hopfield network with neurons possessing a complex-valued discrete activation function. Such memory facilitates natural processing of gray-scale images and has the advantage of mathematical simplicity. The model is based on the concept of multivalued logic and adopts the complex neuron model. Furthermore, sine map is considered as a special one-dimensional unimodal map and has more various phenomenon of bifurca-tions. The coupled sine map networks could be used as a class of associative memory models. And through the discussion on their dynamical behavior, we establish the critical condition on the variance of the fixed points' stability.Due to the indeterministics, unrepeatable and unexpectable properties of the chaotic behavior, more and more researchers tried to associate the chaos properties with stochastic description. From the viewpoint of topology, De-vaney's chaos is undoubtedly considered as the most well-known definition of chaos. Due to the three ingredients of the definition, one may possibly qualify chaotic dynamics in some discretely iterated systems, and even in some concrete infinite systems. By study of the relations between topological transitive, sensitive dependence on initial conditions, and some properties in normal probability theory, we attempt to explore the meanings of chaos more deeply.This thesis is organized as follows: In Chapter 1, we introduce the solid background of the artificial neural networks and their characteristics in the theoretical researches and applications. Furthermore, the research background and progress on the Cohen-Grossberg neural networks, the development of multistate neural associative memory and chaos, are summarized. Also we display the overall structure of the thesis in this chapter.In the second chapter, Gains & Mawhin theorem and some knowledge on the coincidence degree theory are firstly presented. We produce a class of the Cohen-Grossberg neural networks with time delays and high-order terms and several feasible assumptions. By constructing the essential vector function spaces, we prove rigorously the existence of the periodic solutions in this models is proved concretely. Moreover, the networks with the time-variable parameters and delays are investigated as well and the corresponding results are obtained. And, we discretize the continuous models by using the so-called Euler-type scheme, and study the existence of the periodic solutions in thediscrete counterparts in the light of Schauder fixed point theorem.In Chapter 3, we turn to discuss the stability of the periodic solutions in the delayed Cohen-Grossberg neural networks with high-order terms. Sufficient conditions are established for both globally exponential stability and globally asymptotical stability of the existent periodic solution with the aid of the Lyapunov functional and the theory of linear matrix inequality (LMI). Finally, two examples with their numerical simulations are provided to illustrate the possible application of our criteria.In Chapter 4, a kind of the neural networks with practical applications is introduced firstly. Its architecture is builded by fully connected Hopfield network with neurons possessing a complex-valued discrete activation function. By constructing Lyapunov function, we obtain the stability of the network with synchronous dynamics. Moreover, we discuss the existence condition of bifurcation in the case of coupled sine map. The commonness and speciality of the sine map and the logistic map make the sine map have more complex dynamical behavior and more various' bifurcation phenomenon.In Chapter 5, we study the relations between the two famous properties in the Devaney's chaos and the stochastic properties in the research of the complex dynamics in the deterministic system. Here, the large deviations theorem and the central limit theorem are related to the topological transition, and the result is obtained on what stochastic property a specific system should possess whenever its evolution is sensitively dependent on the initial conditions.At the end of this thesis, we review the former results and propose some important topics and prospective work on the delayed neural networks and the on chaotic dynamics.