Positive Invariant Sets And Attractive Sets For Several Time-delays System

As we know, the integration and communication delays are unavoidably encountered both in biological and artificial neural systems. The delays are as double-edged sword to the neural network. On the one hand, the time delays may lead to bad performance to the system such as instability, oscillation, or divergence. On the other hand, the system can benefit from the time delays, such as the synchronizing capacity of complex system can be improved by employing the time delays. Lagrange stability refers to the stability of the total system, while the Lyapunov stability refers to the stability of equilibrium points. And it has been proved that there is no periodic state, bifurcation, and chaos attractor of the system outside positive invariant set and attractive set. Hence, the research on positive invariant set and attractive set of the delay system is necessary and rewarding.In this paper, we study several systems with time delay. Based on the activation function meeting with proper conditions, and by employing Lyapunov functional, delay differential inequality together with linear matrix inequality and so on, we discuss the positive invariant sets and global attractive sets to the several delay system respectively. Meanwhile, compared to the existing references the results obtained in this paper are more general and challenging than that of the existing papers. The remaining paper is organized as follows:In Chapter I, the current advance of study on the systems with time-delay is given, as well as the present development of the relevant research subjects and the main work done in this thesis.In Chapter II, we study the positive invariant set and globally exponentially attractive set for a class of nonlinear separated variables systems with bounded time-varying delays. Based on inequality techniques, the properties of non-negative matrices and vector Lyapunov function method, some algebraic criterions for the the above-mentioned sets are obtained.In Chapter III, the positive invariant sets and global exponential attractive sets for a class of neural networks with unbounded time-delays are studied. Based on assuming that the activation function satisfies the global Lipschitz condition, several algebraic criterions for the aforementioned sets are deserved by constructing proper Lyapunov functions and employing Young inequality.In Chapter IV, we study the global exponential stability in a Lagrange sense for recurrent neural networks with both time-varying delays and general activation functions. Based on assuming that the activation functions are neither bounded nor monotonous or differentiable, several algebraic criterions in linear matrix inequality form for the global exponential stability in a Lagrange sense of the neural networks are obtained by virtue of Lyapunov functions and Halanay delay differential inequality. Meanwhile, the estimations of the globally exponentially attractive sets are given out.In Chapter V, we study the global dissipativity for Cohen-Grossberg neural networks with both time-varying delays and infinite distributed delays. Based on Lyapunov functions and inequality techniques, several algebraic criterions for the global dissipativity are obtained. Meanwhile, the estimation of the positive invariant and globally attractive set are given out.Finally, we sum up all the paper. And the suggestions for further study are given out.