Periodicity and Solution of Rational Recurrence Relation of Order Six

Abstract

Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and ap- plied mathematics. Every dynamical system a n+1 =f(a n ) determines a difference equation and vise versa. We ob-tain in this paper the solution and periodicity of the following difference equation. xn+1=(x n x n-2 x n-4 )/(x n-1 x n-3 x n-5 , (1) n=0,1,... where the initial conditions x -5 ,x -4 ,x -3 ,x -2 ,x -1 and x 0 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.