Analysis Of Long-Time Dynamics For Cellular Neural Networks

The region of cellular neural networks' applications is more and more wide. In some applications where the model of cellular neural network possesses many equilibrium point, it is essential that the network involved is completely stable in the sense that every trajectory converges to some equilibrium point. In some applications, it is desirable that the model possesses a unique and globally asymptotically stable equilibrium point for every external input, especially possesses globally exponentially stable equilibrium point. In some applications, the model necessarily possesses complex dynamical behaviour, e.g, bifurcation and chaos. This thesis mainly studies the existence and number of equilibrium points, the complete stability, global exponential stability, the existence of limit cycles and bifurcations of cellular neural network models. This thesis contains four chapters.In Chapter 1, the background and history, some models and definitions of cellular neural networks which will be studied in this thesis, matrix theory used in this thesis are briefly introduced.In Chapter 2, by using fixed point theory, inequalities, matrix theory and Lyapunov functionals, sufficient conditions of the global exponential stability are obtained on models of standard delayed cellular neural networks with discrete delays, models of standard delayed cellular neural networks with distributed delays, models of delayed cellular neural networks with general output functions respectively. These results are very important in some applications of cellular neural networks, as they ensure that cellular neural networks converge to the unique by exponential speed.In Chapter 3, by constructing Lyapunov functionals and using the essence of the piecewise linearity of the output function and analysis methods, sufficient conditions of the complete stability are given on models of standard cellular neural networks with constand and variable delays respectively.By choosing template parameters as bifurcation parameters, Chapter 4 discusses the global dynamical behaviour of 1-D cellular neural network models, including the number of equilibrium points, the complete stability, the global exponential stability, the existence of limit cycles and bifurcation. These results tell us that cellular neural networks possesse rich dynamical behaviour, and can be used to solve the complex problems.