It is well known that neural networks are complex and large-scale nonlinear dynamical systems. The dynamical characteristics of neural networks include stable, unstable, oscillatory, and chaotic behavior. We must understand the characteristics in order to comprehend profoundly the essence of the practical phenomena, and study deeply the inherent law of the relevant fields. The periodic oscillatory behavior is very important in the differential dynamical systems. Research on it is significant in theory and in practice. The periodic nature of neural impulses is of fundamental significance in the control of regular dynamical functions such as breathing and heart beating. Neural networks involving persistent oscillations such as limit cycle may be applied to pattern recognition and associative memory.In modeling biological or artificial neural networks, it is sometimes necessary to take into account the inherent time delays taking place in the course of the reaction and transmission. More and more researchers have discussed the dynamics of the systems with time delays including the distributed ones and discrete ones. In this theises, the dynamics of neural networks with distributed/discrete delays are discussed respectively. If the time-delay is used as a bifurcation parameter, Hopf bifurcation occurs for these models. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value.We also notice that, the works about Hopf bifurcation usually use the state-space formulation for the differential dynamical systems, referred to as the "time domain" approach. Yet there is another interesting formulation for differential equations in the literature. This alternative representation applies the familiar theory and methodology of feedback engineering systems: an approach described in the "frequency domain", the complex domain after the standard Laplace transforms have been taken in the time domain state-space system. The frequency domain approach was initiated by Allwright, Mees and Chua, Moiola and Chen. This new methodology has the advantage over classical time domain methods. A typical one is its pictorial characteristic that utilizes advanced computer graphical capabilities and so bypasses quite a lot of sophistical mathematical analysis. It is very difficult to analyze the Hopf bifurcation on a neural network with time delays by applying the time domain approach, especially, in the case of a model with distributed delays and the strong kernel. In this thesis, these models are analyzed by means of the frequency domain approach.In this thesis, by means of the frequency domain approach proposed by Moiola, the existence of Hopf bifurcation parameter is determined. If the time delay used as a bifurcation parameter, it is found that Hopf bifurcation occurs when the bifurcation parameter exceeds a critical value. The direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are analyzed by means of the Graphical Hopf Bifurcation Theorem, or according to the direction index and the stability index for the frequency domain approach. The direction index for the frequency domain approach was firstly proposed in our recent paper. |