Behavior and periodic solutions of a two-dimensional systems of rational difference equations

This paper is devoted to investigate the local asymptotic stability, boundedness and periodic solutions of particular cases of the following general system of difference equations: xn+1 = a1yn−1 +a2xn−3 +a3 a4yn−1xn−3 +a5 , yn+1 = b1xn−1 +b2yn−3 +b3 b4xn−1yn−3 +b5 , where the initial conditions x−3, x−2, x−1, x0, y−3, y−2, y−1 and y0 are arbitrary nonzero real numbers and ai and bi for i = 1,2,3,4,5 are arbitrary real numbers. Also, we give some numerical examples to illustrate our results.


Introduction
In the last few decades, there has been great interest in studying difference equation for two reasons.The first reason is difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in physics, ecology, biology, economy, and so on.The second reason is the difference equations can be used in investigating mathematical models which describing real life situations in probability theory, queuing theory, statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical network, quanta in radiation, genetics in biology, economics, psychology, sociology, etc.Recently, there has been great interest in studying difference equations and systems of difference equations of rational form, see ( [1]- [10]) and the references cited therein.Many researchers have investigated the periodic solutions of difference systems for example; in [1], Cinar has determined the positive solution of the rational difference system: Elabbasy et al. [6] has studied the solution of particular cases of the following general system of difference equations: In [8] Elsayed has derived the form of solutions for the rational difference system: , y n+1 = y n−1 ∓1 + x n y n−1 .
Kurbanli [12] considered the behavior of the following system of rational difference equations: [13], Kurbanli et al. discussesd the periodicity of the solutions of the system of difference equations: Also, Kurbanli et al [14] studied the behavior of positive solutions of the system of rational difference equations: Ozban [15] has investigated the solutions of the following system: , y n+1 = by n−3 x n−q y n−q .
Touafek and Elsayed [17] investigated the periodic nature and gave the form of the solutions of the following systems of rational difference equations: In this paper, we discuses the stability character, boundedness and periodic solutions of particular cases of the following general system of difference equations: where the initial conditions x −3 , x −2 , x −1 , x 0 , y −3 , y −2 , y −1 and y 0 are arbitrary nonzero real numbers and a i and b i for i = 1, 2, 3, 4, 5 are arbitrary real numbers.Let us consider eight-dimensional discrete dynamical system of the form: where f : I 4 × I 4 → I and g : J 4 × J 4 → J are continuously differential functions and I, J are some intervals of real numbers.Then for all initial values (x k , y k ) ∈ I × J for k ∈ {−3, −2, −1, 0}, the system of difference equations has a unique solution {(x n , y n )} ∞ n=−3 .Along with the system (1.2) we consider the corresponding vector map: ) is said to be stable if for every ε > 0 there exists δ > 0 such that for every initial conditions where f and g are continuously differential functions at ( − x, − y).The linearized system of (1.2) about the equilibrium point and F J is Jacobian matrix of the system (1.2) about the equilibrium point ( 2 Main results
Theorem 2.1.Suppose that {x n , y n } ∞ n=−3 be a solution of system .Also, assume that x −3 = a , x −2 = b , x −1 = c, x 0 = d, y −3 = A, y −2 = B, y −1 = C and y 0 = D are arbitrary nonzero real numbers such that y −1 x −3 ̸ = 1 and x −1 y −3 ̸ = 1.Then all solutions system (2.3) are periodic with period twenty.Moreover the solutions of the system are: From system (2.3), we see that International Scientific Publications and Consulting Services International Scientific Publications and Consulting Services Hence the solutions are periodic with period twenty.
For the second part of the theorem, the results are holds for n = 0. Now Suppose that n > 0 and our assumption holds for n − 1.That is; International Scientific Publications and Consulting Services
To construct corresponding linearized form of the system (2.4) we consider the following transformation: where f = The Jacobian matrix about the fixed point ( − x, − y) under the transformation (2.5) is given by: which is locally asymptotically stable.
Proof.The equilibrium point of the system (2.4) satisfies the following system of equations: from equations (2.7) and (2.8), the unique equilibrium point is (−1, −1).

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Proof.It follows from system (2.4) that Then the sub-sequences n−0 are decreasing and so bounded from above by M = max{x −.3 , x −.2 x −.1 , x 0 }.From system (2.4) Then the sub-sequences n−0 are decreasing and so bounded from above by M = max{y −.3 , y −.2 y −.1 , y 0 }.

The case:
In this case the system (1.1) reduces to: where the initial conditions x −3 , x −2 , x −1 , x 0 , y −3 , y −2 , y −1 and y 0 are arbitrary nonzero real numbers such that Theorem 2.3.Suppose that {x n , y n } ∞ n=−3 be a solution of system .Also, assume that x −3 = a , x −2 = b , x −1 = c, x 0 = d, y −3 = A, y −2 = B, y −1 = C and y 0 = D are arbitrary nonzero real numbers such that y −1 x −3 ̸ = −1 and x −1 y −3 ̸ = −1.Then all solutions system (2.9) are periodic with period twelve.Moreover the solutions of the system are: International Scientific Publications and Consulting Services Proof.From system (2.9), we see that: International Scientific Publications and Consulting Services Hence the solutions are periodic with period twelve.For the second part of the theorem, the results are holds for n = 0. Now Suppose that n > 0 and our assumption holds for n − 1.That is: Lemma 2.2.Every positive solution of the system (2.9) is bounded.
Proof.It follows from system (2.9) that: Then the sub-sequences n−0 are decreasing and so bounded from above by M = max{x −.3 , x −.2 x −.1 , x 0 }.Also, from system (2.9) Then the sub-sequences

Numerical examples
In this section, we give several interesting numerical examples verifying our theoretical results.This examples represent the periodicity and the stability of solutions of two dimensional systems of rational difference equations (1.1).All plots in this section are drawn with Maple 11.
Example 3.1.Consider the difference system:  Example 3.2.Consider the difference system:    In this work, we studied the local asymptotic stability, boundedness and periodic solutions of particular cases of the general system of difference equations of fourth order (1.1).Also, we gave some numerical examples to illustrate our results.z n , y n+1 = x n x n−1 , z n+1 = 1
Definition 1.1.[10] Let ( − x, − y) be an equilibrium point of the system (1.2).International Scientific Publications and Consulting Services (i) An equilibrium point (

Theorem 1 . 1 .
[16] For the system X n+1 = F(X n ), n = 0, 1, ..., of difference equations such that − X be a fixed point of F. If all eigenvalues of the Jacobin matrix J F about − X lie inside the open unit disk | λ |< 1, then − X is locally asymptotically stable.If one of them has a modulus greater than one, then − X is stable.Definition 1.3.[15] (Periodicity): A sequence {x n } ∞ n=−k is said to be periodic with period p if x n+p = x n for n ≥ −k.

Figure 2 :
Figure 2: Local asymptotic stability of the equilibrium point of System (3.11).