Analysis Of Dynamic Properties Of Difference Equations And Discrete Neural Networks

Modern theory of dynamical systems, which originated at the end of the 19th century with fundamental questions concerning the stability and evolution of the solar system, addresses the long-term behavior of evolving systems. In recent years, there has been a marked increase of research interest in dynamical systems, and the obtained results apply in areas such as physics, biology, meteorology, astronomy, economics, etc. There are two fundamentally different dynamical systems: those with discrete time variable and those with continuous time variable.This dissertation aims at the study of dynamical properties of discrete-time dynamical systems, which include difference equations and discrete neural networks.This dissertation begins with the state-of-the-art of dynamical systems, followed by the fundamental theory that will be used throughout this dissertation. The research results are presented in chapters 3 and 4.Chapter 3 addresses the dynamical properties of difference equations. First, a nonlinear difference equation system is investigated, and it is proved that the positive solutions may be unbounded or converge to a unique equilibrium depending on the values of the system parameters. Then a difference equation with maximum is considered, and some results concerned with the eventually periodicity of the solutions are obtained (especially when the parameters and initial conditions are allowed to be negative).Chapter 4 is intended to study the dynamical properties of a class of discrete neural networks. By introducing the result in the continuous-time neural networks, a discrete-time analogue is formulated by discretizing the continuous-time networks and a sufficient condition is obtained for ensuring the global exponential robust stability of the equilibrium point.Simulation experiments are carried out. Experimental results not only illustrate the dynamical behavior of the solutions, but also validate the presented results.