On the Dynamics of Solutions of Non-linear Recursive System

In this paper, we have studied and examined the periodicity of solutions of rational difference equation system. Then we obtained equilibrium points of this difference equation system and investigated the behaviour of this system related to equilibrium points.


Introduction
In applied mathematics, we can see non-linear difference equations system.Recently, many scientists have interested in many branches of mathematics as well as other sciences with difference equation systems.There have been many investigations and interest in the field of functions of difference equations by several authors.
Nasri et al. introduced a deterministic model for HIV infection in the presence of combination therapy related to difference equations system [2].Grove et al. in [6], studied on the behaviour and existence of the solutions of the rational equation system x n+1 = a .In [7], Cinar and Yalcınkaya examined the periodicity of positive solutions of the difference equation system Clark and Kulenovic,in [1], has investigated the global stability properties and asymptotic behavior of solutions of the recursive sequence , n = 0, 1, 2, . . . .Similar to references above, in this study, we investigated the solutions of following difference equation system Then we obtained equilibrium points of the difference equation systems (1.1) and investigated dynamics of solutions of system (1.1).Now, we give basic, initial definitions and theorems firstly.
Let I 1 and I 2 be some intervals of real numbers and let F 2 : I 1 × I 2 → I 2 be two continuously differentiable functions.Then for every initial condition (x 0 , y 0 ) ∈ I 1 × I 2 , the system of difference equations 3) has a unique solution {x n , y n } ∞ n=0 .Definition 1.1.We say that a solution {x n , y n } ∞ n=0 of the system of difference equations (1.2) and (1.3) is periodic if there exist a positive integer p such that x n+p = x n , y n+p = y n .The smallest such positive integer p is called the prime period of the solution of difference equations (1.2) and ( 1 Definition 1.3.The equilibrium point (x, ȳ) of system equation (1.2) and (1.3) is called stable (or locally stable) if for every > 0, there exists δ > 0 such that for all (x 0 , y 0

Definition 1.4 ([2]
).The equilibrium point (x, ȳ) of system equation (1.2) and (1.3) is called asymptotically stable (or locally asymptotically stable) if it is stable and there exists γ > 0 such that for all (x 0 , y 0 Definition 1.5 ([2]).The equilibrium point (x, ȳ) of system equation (1.2) and (1.3) is called global asymptotically stable if it is stable and for every (x 0 , y 0 ) ∈ ).The equilibrium point (x, ȳ) of system equation (1.2) and (1.3) is called a repeller, if there exists r > 0 such that for all (x 0 , y 0 ) ∈ I 1 × I 2 with 0 < (x 0 , y 0 ) − (x, ȳ) < r, there exists N ≥ 1 such that Lemma 1.7.Consider the quadratic equation Then the following statements hold: (i) A necessary and sufficient condition for both roots of equation

Main results
The obtained results in this section are results in [5].Firstly we give some results related to difference equation system (1.1).In the following theorem, we show the periodicity of this solutions and obtain solutions of system (1.1).
Theorem 2.1.Let {x n , y n } be the solutions of the equation system (1.1) with initial conditions x −1 , x 0 , y −1 , y 0 ∈ R\{0}.Then all solutions of the system (1.1) are periodic with period six.
Proof.From the equation system (1.1),we obtain the following equalities By using Definition 1.1, it is obvious that the solutions are periodic with six period.
It is obvious that theorem holds for n = 0. Assume that it is true the following equalities for n − 1.That is, Therefore, we have to show that it is true for n.Now, we can see that the following results are true by using the equations (2.1) and (1.1) which ends up the induction.Now, in the following theorems, we give the equilibrium points of the equation system (1.1).
Theorem 2.3.If B > 0, then the systems equation (1.1) have two equilibrium points where I 1 and I 2 are some intervals of real numbers.
Proof.Let B > 0. For the equilibrium points of the system equation (1.1), we can write the following equalities from the system (1.1), From above equations, we obtain the following results Theorem 2.4.For the equation system (1.1), the Jacobian matrix is and the characteristic polynomial of the Jacobian matrix J A B , B is also λ 2 +λ = 0 at the equilibrium point (x, ȳ) = ± A B , ± B .

Table 1 .
The stable solutions with 6 period n

Table 2 .
The asymptotically stable solutions with 6 period