Dynamical Properties Of Several Discrete Dynamical Systems

The bifurcation theory of differential equations has many applications in the field of applied mathematics and in life.Bifurcation problems not only exist in continuous dynamical systems,but also exist in discrete systems and have richer dynamical behaviors.This thesis mainly studies the bifurcation problem in two classes of discrete systems,which main contents are as follows:The first chapter mainly summarizes the background,significance and main work of this research.The second chapter briefly reviews an overview of the relevant fundamentals of discrete dynamical systems.In Chapter 3,according to the fact that the discrete version of a known continuous system has not yet been considered in any known literature,we semi-discrete the system to obtain a discrete system,give the parameter conditions of stability of its equilibria,and find some conditions for the supercritical bifurcation and Neimark-Sacker bifurcation to occur in the system.We also use Matlab to carry out numerical simulations to illustrate the conclusions obtained and give the phase diagram,bifurcation diagram,and the Lyapunov diagram respectively.For Chapter 4,the authors in a known literature used the linearization method to obtain several sufficient conditions for the local stability of a positive equilibrium of a known system.And utilizing the iterative scheme and the comparison principle of difference equation,a new sufficient condition to ensure the global asymptotic stability of the positive equilibrium is also obtained.Our main purpose is to consider the dynamical behaviors in its equivalent and simpler form.After semi-discretizing it,we mainly consider local bifurcation problems of the new discrete systems at all equilibrium points,that is,their supercritical bifurcation and flip bifurcation problems,which are more important and difficult.Furthermore,these problems have not been considered in any literature,therefore,our results on these bifurcation problems are completely new.Finally,numerical simulations are given to illustrate the theoretical results obtained.