Dynamics Analysis And Stability Study Of The Neural Network

The neural networks have been extensively studied due to their potential applications in image processing, pattern recognition, optimization problems and many other fields. Understanding the underlying dynamics of these networks is a prerequisite to their practical design and applications. In this thesis, rigorous mathematical analysis is presented for the dynamical behaviors of neural networks. In part I (Chapter 1-4), we address the mono-stability of networks, that is, the existence, uniqueness of a steady-state (equilibrium point, or periodic solution, or almost periodic solution) and its global stability. In part II (Chapter 5-6), we address the multistability of neural networks, that is, the coexistence of multiple steady-states and their local stability.At first, the mono-stability is studied. We investigate a class of Cohen-Grossberg neural networks with both time-varying delays and distributed delays. Under mild conditions, we can prove that any solution is bounded and with some almost periodic properties when t is big enough, which leads to the existence, uniqueness and global stability of the almost periodic solution. We are also concerned with the neural networks with unbounded time-varying delays. Two new concepts, power stability andμ-stability, are proposed. It is revealed that some conditions that guarantee the mono-stability of neural networks with bounded delays can still ensure the mono-stability of neural networks with unbounded delays. And along with the increasing of the delays, the convergence rate is decreasing gradually. As to the high-ordered neural networks, the results reported in the literature are usually derived under the assumption that the activation functions of high-ordered items are bounded by some constants. In this thesis, we remove such assumption and get that there still exists a unique almost periodic solution under some conditions.Secondly, the discussion on multistability of neural networks is another emphasis in our thesis. For a class of piecewise linear activation functions, we can divide the whole state space R~n into some subsets due to their geometrical configurations. Then, rigorous analysis can give that there is a unique equilibrium point in each subset. The attraction basin of each equilibrium point is also estimated. By time reversal, we get that the stable manifolds of the unstable equilibria precisely comprise of the bounds of each attractor in the case of n = 2. We also study the neural networks with almost periodic coefficients and get that there can be multiple almost periodic solutions. Moreover, it shows that the attraction basin of each almost periodic solution is larger than the subset in which it is located. In addition, numerical simulations are presented to illustrate the effectiveness of our results.