Abstract
Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and ap- plied mathematics. Every dynamical system an+1=f(an) determines a difference equation and vise versa. We ob-tain in this paper the solution and periodicity of the following difference equation. xn+1=(xnxn-2xn-4)/(xn-1xn-3xn-5, (1) n=0,1,... where the initial conditions x-5,x-4,x-3,x-2,x-1 and x0 are arbitrary real numbers with x-1,x-3 and x-5 not equal to be zero. On the other hand, we will study the local stability of the solutions of Equation (1). Moreover, we give graphically the behavior of some numerical examples for this difference equation with some initial conditions.
Highlights
Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and applied mathematics
One of the reasons for this is a necessity for some techniques whose can be used in investigating equations arising in mathematical models decribing real life situations in population biology, economic, probability theory, genetics, psychology, ...etc
Difference equations usually describe the evolution of certain phenomenta over the course of time
Summary
Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and applied mathematics. There has been great interest in studying difference equations. There are a lot of interest in studying the global attractivity, boundedness character the periodic nature, and giving the solution of nonlinear difference equations. There has been a lot of interest in studying the boundedness character and the periodic nature of nonlinear difference equations. Difference equations have been studied in various branches of mathematics for a long time. Many researchers have investigated the behavior of the solution of difference equations for examples. We give graphically the behavior of some numerical examples for this difference equation with some initial conditions. Definition (1.2) The difference Equation (2) is said to be persistence if there exist numbers m and M with 0 < m. A sequence x n n k is said to be periodic with period p if xn p xn for all n ≥ −k
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