On a class of nonlinear rational systems of di ff erence equations

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Introduction
Nonlinear difference equations and systems have lately appropriated the interest of numerous researchers. In fact, these types of equations have various implementation not just in mathematics but in describing some natural life phenomena, which that appear in engineering, economics, ecology, and so on. This can be due to the reality that these phenomena can be represented as modelled by using systems of difference equations. Wherefore, it has enticed the attention of a huge number of scholars and researchers over the past few years. Lately, there has been a great number of published papers growing constantly in this scope. From amid of many newly published papers in this area, we can present some of these studied as following: In [1] Elsayed et al. have got the solutions expressions for the following nonlinear system of difference equations R n+1 = a 1 T n−1 S n−1 R n−1 + S n−1 + T n−1 , S n+1 = a 2 T n−1 R n−1 R n−1 + S n−1 + T n−1 , T n+1 = a 3 R n−1 S n−1 R n−1 + S n−1 + T n−1 .
Also, the authors in [2] obtained the forms of the solutions and periodic nature of the following rational systems of difference equations x n+1 = y n−1 z n z n ± x n−2 , y n+1 = z n−1 x n x n ± y n−2 , z n+1 = x n−1 y n y n ± z n−2 .
El-Dessoky et al. [3], built the form of the solutions of the following nonlinear systems z n+1 = z n t n−1 ±t n ± t n−1 , t n+1 = t n z n−1 ±z n ± z n−1 .
The form of the solutions of the following third order systems x n+1 = y n z n−1 y n ± x n−2 , y n+1 = z n x n−1 z n ± y n−2 , z n+1 = x n y n−1 x n ± z n−2 , were obtained by Alayachi et al. [6]. Moreover, in [7] the authors have explored the structure solutions of the following fifth order systems of recursive equations x n+1 = x n−3 y n−4 y n (1 + x n−1 y n−2 x n−3 y n−4 ) , y n+1 = y n−3 x n−4 x n (±1 ± y n−1 x n−2 y n−3 x n−4 ) .
Abdulkhaliq and Shoaib in [8] highlighted on the dynamics of the systems Also, Din [9] investigated the behaviour solutions of a Lotka-Volterra model x n+1 = αx n − βx n y n 1 + γx n , y n+1 = δy n + ϵ x n y n 1 + ηy n .
Finally, the forms and expressions of the solution of the difference equations systems x n+1 = y n y n−2 ±x n−3 ± y n−2 , y n+1 = x n x n−2 ±y n−3 ± x n−2 has been founded by authors in [10].
For more interesting papers on this scope can be seen in [11][12][13][14][15][16][17][18][19][20][21]. More precisely, the forms of exact solutions of some models of nonlinear systems of difference equations cannot be sometimes extracted. Therefore, the explore of exact solutions and other behaviors of a higher order of nonlinear systems of difference equations is quite challenging and valuable due to the importance of its applications.
Motivated by the remindered studies above , the major purpose of this paper is to construct a general form of the solutions and periodicity of the model of each systems, and so our main contributions in this regard involves: • Obtaining the forms of the exact solutions of each systems by using manual iterations to get the final formula of the solutions. • Investigate the periodicity of the solution of each systems of difference equations. • Using Fibonacci sequence to formulate the exact solutions of some systems.
• Confirming our theoretical results graphically and obtain the numerical results to explain the behaviours of the solutions by using some mathematical programming such as MATLAB.
From system (2.1), we have Other formulas can be obtain by similar way. Thus, the proof is complete.

Next, from system (2.2) we have
Other formulas can be proved by identical way. Thus, this completes our proof.
2.3. On the system: S n+1 = T n S n−2 S n−2 + T n−1 , T n+1 = S n T n−2 −T n−2 + S n−1 This section aims to present a fundamental theorem that states the existence of periodic twelve solutions of the following system: s n+1 = t n s n−2 s n−2 + t n−1 , t n+1 = s n t n−2 −t n−2 + s n−1 , with positive real numbers initial conditions s −2 , s −1 , s 0 , t −2 , t −1 , t 0 . Theorem 3. Let {s n , t n } ∞ n=−2 be a solution of system (2.3). Then the solutions of this system have a periodic solutions with period twelve and for n = 0, 1, ..., Proof. Obviously solutions true if n = 0. Now, let n > 0 and assume that our solutions are satisfied for n − 1. That is; , , t 12n−3 = cd b + d .

Next, system (2.3) follows that
By similar technique, we can show the other relations. Hence, the is end our proof.
2.4. On the system: S n+1 = T n S n−2 S n−2 + T n−1 , T n+1 = − S n T n−2 T n−2 + S n−1 Here, our principal task is to formulate the solutions of the following system of difference equations: s n+1 = t n s n−2 s n−2 + t n−1 , t n+1 = − s n t n−2 t n−2 + s n−1 , where s −2 , s −1 , s 0 , t −2 , t −1 , t 0 are defined as positive real numbers.
Also, we found an exact solutions' forms of the system S n+1 = T n S n−2 S n−2 + T n−1 , T n+1 = S n T n−2 T n−2 − S n−1 which we described and proved each iteration in Theorem (2.2). In section (2.3), we created a fundamental theorem (2.3) that states the existence of periodic twelve solutions of system S n+1 = T n S n−2 S n−2 + T n−1 , T n+1 = S n T n−2 −T n−2 + S n−1 . We have also explored existence of shape and periodic of the solutions of system S n+1 = T n S n−2 S n−2 + T n−1 , T n+1 = − S n T n−2 T n−2 + S n−1 in section (2.4). Finally, in section (3) we confirmed our theoretical results in the previous sections by carried out numerical simulation using MATLAB programm.