Dynamics Analysis For Several Classes Of Neural Network Models

In the last decades, neural networks have been extensively studied and applied in many different fields such as associative memory, signal and image processing, solving nonlinear algebraic equations, pattern recognition and some optimization problems. In such applications, it is of prime importance to ensure that the de-signed neural networks are stable. In practice, due to the finite speeds of the switching and transmission of signals, time delays do exist in a working network. On the other hand, time delay is unavoidable in many evolution processes. Mean-while, the state of electronic networks may experience abrupt change at certain moment of time, i.e., the impulse effect always exist in neural networks. Thus, it is necessary to study neural networks with time delays and impulses. In addition to the delay effects, recently, studies have been intensively focused on stochastic mod-els, it has been realized that the synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other prob-abilistic causes, and it is of great significance to consider stochastic effects on the stability of neural networks or dynamical system described by stochastic functional differential equations.In this thesis, we describe some important properties of the dynamic behaviors of several class of neural networks model, which includes stability, pth moment exponential stability, periodic solution and bifurcation, ω—limit set, initial valued problem and equilibrium points.We study the p-th moment exponential stability of fuzzy cellular neural net-works with time-varying delays under impulsive perturbations and stochastic noises. Based on Lyapunov function, stochastic analysis and differential inequality tech-nique, a set of novel sufficient conditions on p-th moment exponential stability of the system are derived without the assumption of the differentiability and mono-tonicity of time delays function. These results generalize and improve some of the existing ones. Moreover an illustrative example is given to demonstrate the effectiveness of the results obtained.The problem of existence of periodic solutions and ω—limit set in a class of discrete-time cellular neural networks is another key research topic in this thesis. It is well-known that the discrete-time cellular neural networks were first proposed be H. Harrer and J. A. Nossek in1992. To neural networks, discrete analogues of continuous-time models are very important for the theoretical analysis and the implementation of neural networks. Many phenomena are described by discrete-time systems and in the fields of engineering especially the numerical simulation, the models always are discrete-time systems.We investigate a class of discrete-time model of two-cell cellular neural net-works with γ—symmetric. By using the analytical technique to construct return mappings, and based on fixed point theorem of return mappings and γ—sym-metric of systems, a complete analysis is provided including existence of periodic solutions, ω—imit set and global attractor. Our analysis shows that such discrete-time cellular neural networks have periodic solutions and closed invariant, and their solutions, except for fixed points, eventually stay on the closed curves. We observe an interesting distribution of periodic solution, ω—limit set and global attractor of systems.Differential equations with discontinuous right hand-side provide mathemat-ical models for many applications in science and engineering. We mention here mechanical systems with nonsmooth harmonic oscillator, neural networks with dis-continuous neuron activations, mechanical systems with Coulomb friction, valve oscillators with a discontinuous characteristic, and so on. In this thesis, we de-scribe some important properties of the dynamic behaviors of a class of neural networks model with discontinuous activations. Using the theory of differential inclusions and the method of the corresponding Poincare map of flow and extend-ing Poincare-Bendixson theorem, some interesting results are provided including uniqueness and non-uniqueness for the initial-value problem, existence of sliding motion solutions, stability and geometrical properties of equilibrium points, num-ber and stability of periodic trajectories, especially close trajectories with sliding and homoclinic trajectories with sliding. Synchronously we give necessary condi-tion for the existence of closed trajectories in our different cases. In the study of closed trajectories, some interesting dynamical behaviors are found. In addition to limit cycles of the system, closed trajectories are present in the form of periodic trajectories. In particular, it also comes about that all solutions of the system are periodic trajectories.