The Research On Dynamical Behaviour Of Some Difference Equations

This dissertation aims to study the global asymptotical stability of the nonlinear rational difference equations, which are some"Open Problems and Conjectures"given by Ladas, et al. On the basis of the previous conclusions, we in detail discuss the dynamical behavior of nonlinear difference equations by using the stability theory, convergence theorem,the method of semicycle analysis and other techniques. Namely, the length of the semicycle, the existence of periodic two solutions and non-oscillatory solutions, the stability, global attractivity, loacl or global asymptotical stability of the positive equilibrium.The content of the dissertation is divided into four chapters, namely:Chapter 1, first, we simply introduce the historical background and the progress of the difference equations, give the origin of the difference equation discussed in this article. Second, we list the concepts, the theorems and the known conclusions which are used in this article.Chapter 2, the following third-order nonlinear difference equation is considered in detail, where b∈[0,1]and the initial values x? 2 , x? 1 , x0∈[0,∞).We thoroughly consider the the rule of the lengths of positive and negative semicycles of this equation by utilizing the "Semicycle Analysis Method".Resultly, its positive equilibrium point is verified to be globally asymptotically stable.Chapter 3, we investigate the global behavior of all non-negative solutions for the rational difference equation Where the parameters p , q≥0, r > 0,kis a positive integer,and the initial conditions x? k, x? 1 ,x0 are non-negative real numbers, mainly for the existence and the non-existence of eventual non-negative prime period-two solutions, the global attractivity and the global asymptotic stability for its equilibria points to its all positive solutions and the existence and asymptotic behavior of non-oscillatory solutions.Chapter 4, using the methods similar to that in Chapter 3, we investigate the global behavior of all non-negative solutions for the rational difference equation where p , q≥0, r > 0,kis a positive integer,and the initial conditions x? k, x? 1 ,x0are nonnegative real numbers. Our results show that the positive equilibrium point of this equation is a global attractor under appropriate conditions and that the zeroequilibrium of this equation is globally asymptotically stable.