On a System of Two High-Order Nonlinear Difference Equations

f(x n , x n−1 , . . . , x n−k ) determines a difference equation and vice versa. Recently, there has been great interest in studying the system of difference equations. One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real life situations in population biology, economic, probability theory, genetics psychology, and so forth. The theory of difference equations occupies a central position in applicable analysis. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. Nonlinear difference equations of order greater than one are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points. The study of properties of rational difference equations and systems of rational difference equations has been an area of interest in recent years. There are many papers in which systems of difference equations have been studied. Çinar et al. [1] have obtained the positive solution of the difference equation system:


Introduction
Difference equations or discrete dynamical systems are diverse fields which impact almost every branch of pure and applied mathematics.Every dynamical system  +1 = (  ,  −1 , . . .,  − ) determines a difference equation and vice versa.Recently, there has been great interest in studying the system of difference equations.One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real life situations in population biology, economic, probability theory, genetics psychology, and so forth.The theory of difference equations occupies a central position in applicable analysis.There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole.Nonlinear difference equations of order greater than one are of paramount importance in applications.Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering, and economics.It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points.
The study of properties of rational difference equations and systems of rational difference equations has been an area of interest in recent years.There are many papers in which systems of difference equations have been studied.
C ¸inar et al. [1] have obtained the positive solution of the difference equation system: C ¸inar [2] has obtained the positive solution of the difference equation system: Also, C ¸inar and Yalc ¸inkaya [3] have obtained the positive solution of the difference equation system: Özban [4] has investigated the positive solutions of the system of rational difference equations: Advances in Mathematical Physics Papaschinopoulos and Schinas [5] investigated the global behavior for a system of the following two nonlinear difference equations: where  is a positive real number, ,  are positive integers, and  − , . . .,  0 ,  − , . . .,  0 are positive real numbers.Clark et al. [6,7] investigated the system of rational difference equations: where , , ,  ∈ (0, ∞) and the initial conditions  0 and  0 are arbitrary nonnegative numbers.
Ibrahim [9] has obtained the positive solution of the difference equation system in the modeling competitive populations: Din et al. [10] studied the global behavior of positive solution to the fourth-order rational difference equations: where the parameters , , ,  1 ,  1 ,  1 and the initial conditions  − ,  − ,  = 0, 1, 2, 3 are positive real numbers.Although difference equations are sometimes very simple in their forms, they are extremely difficult to understand thoroughly the behavior of their solutions.In [11], Kocić and Ladas have studied global behavior of nonlinear difference equations of higher order.Similar nonlinear systems of rational difference equations were investigated (see [12,13]).Other related results reader can refer to [14][15][16][17][18][19][20][21][22].
(iv) (, ) is unstable if it is not stable.
Theorem 3 (see [11]).Assume that  +1 = (  ),  = 0, 1, . .., is a system of difference equations and  is the equilibrium point of this system; that is, () = .If all eigenvalues of the Jacobian matrix   , evaluated at , lie inside the open unit disk || < 1, then  is locally asymptotically stable.If one of them has modulus greater than one, then  is unstable.

Then all roots of the polynomial 𝑝(𝜆) lie inside the open unit disk |𝜆| < 1 if and only if
where Δ  is the principal minor of order  of the  ×  matrix:
We summarize the local stability of the equilibria of (10) as follows.
The following theorem is similar to Theorem 3.4 of [8].
Theorem 8. Let (  ,   ) be positive solution of system (10), then for  ≥ 0 the following results hold: Proof.It is true for  = 0. Suppose that results are true for  = ℎ ≥ 1, namely, Now, for  = ℎ + 1, by virtue of system (10), we have and similarly, and similarly, Hence, for ∀ ≥ 0, the results are true.
Proof.From (i) of Theorem 5, we obtain that the unique equilibrium point (0, 0) of system (10)

Rate of Convergence
In this section we will determine the rate of convergence of a solution that converges to the equilibrium point (0, 0) of the system (10).The following result gives the rate of convergence of solution of a system of difference equations: where   is an -dimensional vector,  ∈  × is a constant matrix, and  :  + →  × is a matrix function satisfying where ‖ ⋅ ‖ denotes any matrix norm which is associated with the vector norm.
Theorem 10 (see [23]).Assume that condition (31) holds, if   is a solution of (30), then either   = 0 for all large  or exists and is equal to the modulus of one the eigenvalues of the matrix .
Assume that lim  → ∞   = , lim  → ∞   = , we will find a system of limiting equations for the system (10).The error terms are given as Theorem 11.Assume that  > 1,  > 1, and {(  ,   } are a positive solution of the system (10).Then, the error vector   of every solution of (10) where   (0, 0) is equal to the modulus of one the eigenvalues of the Jacobian matrix evaluated at the equilibrium (0, 0).

Numerical Examples
In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section.These examples represent different types of qualitative behavior of solutions to system of nonlinear difference equations. 0 = 1.6, moreover, choosing the parameters  = 0.8,  = 0.9 and  = 5.Then system (10) can be written as The plot of system (43) is shown in Figure 2.

Conclusions and Future Work
In this paper, we discussed the dynamics of high-order discrete system which is extension of [8,10,14].We conclude that (i) the equilibrium point (0, 0) is globally asymptotically stable if  > 1,  > 1, (ii) the equilibrium (0, 0) and ( 1+ √1 − , 1+ √1 − ) if  < 1 or  < 1 is unstable.Some numerical examples are provided to support our theoretical results.It is our future work to study the dynamical behavior of system (10) when  = 1 or  = 1.