Rational Difference Equations And Their Approximate Equation For The First Time The Stability Study

In many cases,time evolving variables are discrete.Difference equations appear as natural descriptions of evolution phenomena,especially under the rapid development of computer science and technology.Recently,difference equations have been used in various fields such as numerical analysis,control theory and computer science etc.Some mathematicians,such as S.A.Kuruklis and G.Ladas,have many good results about stability of linear and rational difference equations.They present many open problems and conjectures about some interesting types of difference equations.This paper devoted to the asymptotic stability of two classes of 4th order delay difference equations.One of important methods for discussing stability of linear difference equations is to study the distribution of all the roots of our characteristic equation(polynomial equation)in the complex plane.When the degree of our polynomial equation is greater or equal five,we can not use normal methods or mathematical softwares to get formula solutions.In this paper,by using the method of characteristic roots and other methods,we study the distribution of all characteristics roots when the parameters l is even and odd in chapter 2,and give necessary and sufficient conditions for asymptotic stability of the zero solution of delay difference equations x_n+4-ax_n+bx_n-l=0.Finally this paper,on this basis and using known lemma,gives criterion for the stability of the zero solution of rational difference equations x_n+1=(β₃x_n-3+β_l+3x_n-l-3)/(A+B₃x_n-3+B_l+3x_n-l-3).It provides a theoretical basis for solving the practical problems with such model.