Stability And Bifurcation Analysis Of Neural Network

Neuron is the basic unit of structure and function of the nervous system, which has complex nonlinear characteristics. The discharge activities of neuron often exhibit rich varieties of dynamical behaviors, such as stability, bifurcation and Chaos. So it is particularly important to study on the stability and bifurcation of neurons. In addition, the neural network consists of thousands on thousands of neurons linked by different methods. Neural networks with delays, as one of the basic equations which can describe the motion of nature, usually solve the non‐linear problems and have rich varieties of dynamical behaviors, in which the stability, bifurcation and periodic motion are the basic features, which has been successfully applied to the optimization calculation, associative memory and pattern recognition. Therefore, it is realistic meaning to study dynamics of neural networks with delays. In this thesis, based on the analysis and summary of research status of reaction‐diffusion systems, employing partial functional differential equation theory, stability, bifurcation and Turing’s theory, the author investigates stability and bifurcation of reaction‐diffusion neural networks with delays. In addition, application of fractional calculus theory to discuss the stability problem of fractional‐order neuron system. The organization of this paper takes the following form:In the first Chapter of this dissertation, the author elaborate the current status and research progress about reaction‐diffusion neural network systems and expounds the main contents of this paper.The second Chapter studies2‐D reaction‐diffusion neural network system with delays, presenting some sufficient conditions ensuring the equilibrium of system to be stable and derive conditions on the parameters so that spatial homogenous Hopf bifurcation and Turing instability occur.The third Chapter deals with the stability and Hopf bifurcation of two classes of neural networks with time‐lags and reaction‐diffusion, determining as the bifurcation parameter respectively, give some criteria for stability and Hopf bifurcation.The fourth Chapter discusses the stability and instability of H‐R neuron model with fractional orders, give some criteria for stability and instability.The fifth Chapter summarizes the research work of this dissertation. Furthermore, the future research direction is made.