Order Linear Delay Differential Equation Stability

Difference equations are also called discrete dynamic systems, which arise from the researches on discretization formats of differential equations. At present, study of higher order difference equations mainly concentrates on their stability and actual utility, which becomes one of the focuses of researches of difference equation theory. The paper studies stability of a class of delay difference equations and establishes necessary and sufficient conditions for its zero solution to be asymptotically stable. By using the method of characteristic roots and other methods, we see that how to find the parameter field of asymptotic stability for the above equations is the key and difficulty.The paper is organized as follows.In chapter 1, we introduce the background and the present situation of researches of difference equations, and the main content and results of this paper. In chapter 2, we locate the situation of the characteristic roots of 8-th order difference equations of the form x_n+8-ax_n+bx_n-k=0. In chapter 3, we obtain necessary and sufficient conditions for the zero solution of the above equation to be asymptotically stable by using the method of the characteristic roots.