Periodicity and Solution of Rational Recurrence Relation of Order Six

Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and applied mathematics. Every dynamical system   1 n a f a   n determines a difference equation and vise versa. We obtain in this paper the solution and periodicity of the following difference equation.     1 2 4 1 3 n n n n n n n x x x x x x x       5  , (1) 0,1, n   where the initial conditions 1 5 4 3 2 , , , , x x x x x      and 0 x are arbitrary real numbers with 1 3 , x x   and 5 x 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.


Introduction
Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and applied mathematics.Every dynamical system  1 n n determines a difference equation and vise versa.Recently, there has been great interest in studying difference equations.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.Recently there are a lot of interest in studying the global attractivity, boundedness character the periodic nature, and giving the solution of nonlinear difference equations.Recently 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.First results in qualitative theory of such systems were obtained by Poincaré and Perron in the end of nineteenth and the beginning of twentieth centuries.For some results in this area, see for example [1][2][3][4][5][6][7][8][9][10][11][12][13].
Although difference equations are sometimes very simple in their forms, they are extremely difficult to un-derstand throughly the behavior of their solutions.
Many researchers have investigated the behavior of the solution of difference equations for examples.
Cinar [1,2] investigated the solutions of the following difference equations gave that the solution of the difference equation   G. Ladas, M. Kulenovic et al. [12] have studied period two solutions of the difference equation [13] obtained the solution of the difference equation Ibrahim [5] studied the third order rational difference Equation In this paper we obtain the solution and study the periodicity of the following difference equation 0,1, n   where the initial conditions 5 , x  , and 0 x are arbitrary real numbers with 1 x  , 3 x  and 5 x  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.
Here, we recall some notations and results which will be useful in our investigation.
Let I be some interval of real numbers and Let be a continuously differentiable function.Then for every set of initial conditions 1 , , has a unique solution [11].
2) The difference Equation ( 2) is said to be persistence if there exist numbers m and M with 0 < m ≤ M < ∞ such that for any initial    there exists a positive integer N which depends on the initial conditions such that m ≤ x n ≤ M for all n ≥ N.

Definition (1.3) (Stability)
Let I be some interval of real numbers.
1) The equilibrium point x of Equation ( 2) is locally stable if for every ε > 0, there exists δ > 0 such that for 1 0 , , , 2) The equilibrium point x of Equation ( 2) is locally asymptotically stable if x is locally stable solution of Equation (2) and there exists γ > 0, such that for all 1 0 , , , The equilibrium point x of Equation ( 2) is globally asymptotically stable if x is locally stable, and x is also a global attractor of Equation (2).
5) The equilibrium point x of Equation ( 2) is unstable if x not locally stable.
The linearized equation of Equation (2) about the equi-librium x is the linear difference equation Theorem (1.4) [10] Assume that (real numbers) and is a sufficient condition for the asymptotic stability of the difference equation 1 0, 0,1, .
Remark (1.5) Theorem (1.4) can be easily extended to a general linear equations of the form where 1 2 , , , k p p p    (real numbers) and for all n ≥ −k.

Solution and Periodicity
In this section we give a specific form of the solutions of the difference Equation (1).

Proof:
For n = 0 the result holds.Now suppose that n > 0 and that our assumption holds for n − 1.We shall show that the result holds for n.By using our assumption for n − 1, we have the following:     Now, it follows from Equation (1) that           14 5 14 6 14 8 14 10 14 7 14 9 14 11 , , , .
Thus, the proof is completed.

Theorem (2.2)
Suppose that be a solution of Equation ( 1).
Then all solutions of Equation ( 1) are periodic with period fourteen.
 

Proof:
From Equation (1), we see that which completes the proof.

Stability of Solutions
In this section we study the local stability of the solutions of Equation (1).

Proof:
For the equilibrium points of Equation ( 1), we can write   Thus the equilibrium points of Equation ( 1) is are 0 and 1.

Theorem (3.2)
The equilibrium points 0 x  and 1 x  are unstable.

Proof:
We will prove the theorem at the equilibrium point 1 x  and the proof at the equilibrium point 0 x  by the same way.
Let be a continuous function defined by Then the linearized equation of Equation ( 1 By the generalization of theorem (1.4) we have which is impossible.This means that the equilibrium point 1 x  is unstable.Similarly, we can see that the equilibrium point 0 x  is unstable.

Numerical Examples
For confirming the results of this section, we consider numerical examples which represent different types of solutions to Equation (1).

Acknowledgements
We want to thank the referee for his useful suggestions.

Figure 1 .
Figure 1.The periodicity of solutions with period 14 with unstable equilibrium points x = 1 and x = 0.

Figure 3 .
Figure 3. Periodicity of solutions with period 14 with unstable equilibrium points x = 1 and x = 0.