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. 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. The dynamics of neural networks with time delays have been discussed by many researchers. It is well known that a neural network model with distributed delay is more general than one with discrete delay. In this paper, a more general two-neuron model with distributed delays is investigated. We notice that, the works about Hopf bifurcation usually use the state-space formulation for delayed differential equations, referred to as the "time domain" approach. Yet there is another interesting formulation for the 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 [1], Mees and Chua [2], Moiola and Chen [3, 4]. 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 two-neuron model with distributed delays and the strong kernel. In this paper, these models are analyzed by means of the frequency domain approach.In this paper, by means of the frequency domain approach proposed in [5], theexistence of Hopf bifurcation parameter is determined. If the mean 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. Some numerical simulation results and the frequency-domain graphs are given, verifying the theoretical analysis results.In Chapter one, we firstly give a briefing of some fundamental mathematical concepts and results of nonlinear dynamical systems, then both the time domain and the frequency domain approach to the classical Hopf bifurcation theorem will be introduced. Finally the advantages of the frequency-domain approach over the classical time-domain approach are addressed. In Chapter two, a general two-neuron model with distributed delays for a weak kernel / strong kernel is investigated. By introducing a "state-feedback control", one obtains a linear system with a nonlinear feedback. The mean delay is used as a bifurcation parameter. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter for the model is proven. The algebraic equations for computing the Hopf bifurcation are obtained.In Chapter three, the direction and stability of the bifurcating periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. The curvature coefficient will be introduced for determining the direction of the Hopf bifurcation. By drawing the frequency graph of the eigenvalue-locus and the half-line L1, one can determine the stability of the bifurcating periodic solution.In Chapter four, some numerical simulation results and the frequency graph are presented to justify the theoretical analysis results. |