Existence And Stability Of Periodic Solutions Of Neural Networks With Impulses And Delays

In many fields of modern science and technology, impulses or delays are now in widespread use. Compared with the differential equation without impulses and delays, the impulsive differential equation and the time-delay differential equation can reflect the real developing process, its prominent characteristic is that they can fully consider the influence caused by the instantaneous sudden change and the time delay, so that we can investigate the change rule of the thing precisely. As we all known, delays and impulses have great influence on the stability of the neural network. Specially, some neural networks without delays or impulses are stable, however, the stability of the original system would be unstable if the neural networks were introduced to delays or impulses. Therefore the structure of the system will be changed in essence, simultaneously the stability analysis of the neural networks with delays will become more difficult after the introduction of impulses.In this paper, by using Mawhin's coincidence degree theory, bifurcation method and the numerical simulation, several kinds of neural networks with impulses and/or delays are investigated.Chapter 1 (Introduction) gives an introduction to some basic concepts. Such as the significance and the application of neural network's research, impulsive differential equation, Lyapunov function, stability definition and stability theory of differential equation and chaotic strange attractor.In Chapter 2, by using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions, we study the existence, uniqueness and global exponential stability of periodic solutions for nonautonomous cellular neural networks with impulses and delays. Further, using the numerical simulation method the chaotic property of the system is investigated. The results generalize the previous model without impulses.In Chapter 3, by using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions, the global exponential stability and periodicity are investigated for a class of delayed high-order Hopfield neural networks with impulses, which are new and complement previously known results.In Chapter 4, we consider systems of delay differential equations representing the models containing two cells with any time-delayed connections. Global stability, delay-independent and delay-dependent local stability are studied, and the condition for absolute local stability is given.Notably, in Chapter 3 and 4, by using the numerical simulation, we found new Gui strange attractors which are caused by impulsive perturbations. They are different from the Lorenz attractor and the Rossler attractor.In the last Chapter, we summarize the content of the article, and go along expecting discussion.