The Stabilityand Hopf Bifurcation Of Complex Networks With Time Delay

Stability and bifurcation of complex networks is one of the important and difficult study for network dynamics behavior, and the study of neural networks has a very important theoretical and practical significance. I select several different common neural network models as the research object, and aim at analyzing the stability and Hopf bifurcation of the neural networks with time-delay, and discussing the direction and stability of the bifurcating periodic solutions. This thesis combines with the characteristics of root analysis method, and the mainly investigation and study theories here are the theory of normal form and the center manifold theorem.This thesis separately analyzes the network stability and Hopf bifurcation in three different neural networks with time-delay in detail. The main research contents and innovation in this thesis can be summarized as follows:In the third chapter, an annular neural network model with three neurons and a delay is considered. When leading axonal transmission signal in the annular neural network model with three neurons, the corresponding dynamic system becomes a nonlinear dynamic system with delays, and the dynamic properties will become very complex. The dynamic behavior perhaps evolve to stable equilibrium point, produce periodic oscillation or chaos. So this chapter take time-delay as the bifurcation parameter to analyze the stability and Hopf bifurcation at the equilibrium point.The fourth chapter mainly consider an yuan an n-neuron annular neural model.Annular neural network exists in such as cerebral cortex, cerebellum, and hippocampus and even in the chemical medium and circuit design. And for the high dimensiondynamics model can better reflect properties of neural network, so this chapter studies an n-neuron annular neural model and take the sum of delay1 2 3 n?=?+?+?+?+? as bifurcation parameter to analyze the stability and Hopf bifurcation, and discuss the direction of Hopf bifurcation and the stability of periodic solutions.The dynamics of a general two-neuron model with distributed delays are investigated in the fifth chapter. It mainly take rate ? as bifurcation parameter to analyze the conditions in detail which the Hopf bifurcation occurs at the equilibrium for the strong kernel(2), 0(sF s se??????). Besides, it discuss the direction of Hopf bifurcation and stability of periodic solutions.