Research On Some Dynamical Behavior Of Delay Dynamic Systems

With the increasing development of science and technology, many mathematical models which are described by delay dynamical equations(delay dynamical systems)are proposed in the field of natural science and edging field including physics, mechanics, control theory, biology, medicine and economics. Therefore, the study of delay dynamical systems is important in theory and practical application.Delay dynamical systems belong to nonlinear dynamics. Classic nonlinear dynamics focus on weakly nonlinear and weakly coupled systems by the methods of disturbance and asymptotic analysis. While modern nonlinear dynamics is different from classic one, it studies the rules of the qualitative and quantitative change of systems. The research methods are precision. The systems are strongly nonlinear. The research objectives are stability, periodic solution, attractor, bifurcation, chaos, and solitons. The aim is to discovery the complexity of nonlinear dynamics.In this paper, the author studies further on some mathematics models.Considering the effecting of time delay to the stability of the systems and applying many methods, the author investigates global dynamical behaviors of several kinds delay functional differential equations. The author studys the existence of periodic solutions, robust stability, bifurcation, and introduce the solution of functional differential equations and neural network simulation with Matlab.This paper is composed of six chapters and main results are described as follows: First, the author introduces the historical background and the recent development of delay dynamic systems and neural networks. Moreover, the main results of this paper are also briefly introduced.Second, by applying coincidence degree theory,the existence of periodic solution of Duffing-type and generalized Liénard-type equation with delay is investigated, and some sufficient criteria are established,which improve and generalize some known results.Third, one hand, applying topological degree theory and constructing suitable Liapunov functional, the author investigates asymptotic robust stability of interval cellular neural networks with S-type distributed delays on finite intervals. The other hand, on the base of studying of continuity and differentiability of Lebesgue-Stieltjes integration on infinitely interval, the author investigates global exponential robust stability of interval cellular networks with S-type distributed time delay by applying the new analysis techniques and constructing suitable Liapunov functional with Lebesgue -Stieltjes integration. In two cases, the convenient criteria and examples are presented. The conclusions are useful to analyze the stability of systems.Fourth, via the intercept functions and intercept equation, the author presents some sufficient conditions for the existence of a unique globally exponentially stable equilibrium of reaction-diffusion cellular neural networks (RDCNNs) with S-type distributed time delay, in which the boundedness, monotonicity and differentiability of signal functions f j and g j,j = 1,2,,n defined on R are not required. The results are verifiable and the method of M-matrix is more practical.Fifth, the associative memory of neurons model with nonmonotone dynamics is considered, where the output function is not sigmoid but nonmonotonic. After analyze the bifurcating phase plot of the nonmonotonic system, the author obtains the asymptotically stability criteria based on the characteristic equation and technique in Hassard. The stability and direction of the Hopf bifurcation are studied by applying the normal form theory and the center manifold theorem. Finally, numerical simulations have been used to demonstrate the change of phase plot with respect to the time delay. The networks studied above have value in application to some extend.Finally, the studying contents and the main results of this paper are briefly summarized, furthermore, some research prospects are also proposed.