On the solutions of a rational system of difference equations

Document Type : Original Article

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The high institute for Engineering & Technology, Al-Obour

Abstract

Difference equations is one of the fundamental topics of mathematics that is the used to understand the behavior of models defined in discrete time domain. Difference equations are used to approximate differential equations. The history of difference equations dates back to very old times. Although they are seen as discrete structures of differential equations, they have a much older history.
In this paper, we study the admissible solutions of the system of difference equations
x_(n+1)=x_n/y_n , y_(n+1)=x_n/(ax_n+by_n ), n=0,1,…,
where a,b are nonnegative real number such that (a+b≠0) and the initial values x_0, y_0 are nonzero real numbers. We show that the equilibrium point (b/(1-a),1) of the abovementioned system is locally asymptotically stable when
|a|<1
We show also that the equilibrium point (b/(1-a),1) is globahly asymptotically stable.
When |a|>1, the equilibrium point is unstable (saddle point) . and finally, it is nonhyperbolic point when |a|=1.
We shall also introduce the forbidden set and provided some illustrative examples.

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