Qualitative study of a third order rational system of difference equations

This paper is concerned with the dynamics of positive solutions for a system of rational difference equations of the following form un+1 = αun−1 β + γvn−2 , vn+1 = α1v 2 n−1 β1 + γ1un−2 , n = 0, 1, . . . , where the parameters α, β, γ, α1, β1, γ1 and the initial values u−i, v−i ∈ (0,∞), i = 0, 1, 2. Moreover, the rate of convergence of a solution that converges to the zero equilibrium of the system is discussed. Finally, some numerical examples are given to demonstrate the effectiveness of the results obtained.

To the best of our knowledge, the difference equations system (2) was not dealt with. Therefore, it is meaningful to study their deep results.

Preliminaries
For the completeness in this paper, we find useful to remind some basic concepts of the difference equations theory as follows: Let f 1 : I 3 1 × I 3 2 → I 1 and f 2 : I 3 1 × I 3 2 → I 2 are continuously differentiable functions where I 1 , I 2 are intervals of real numbers. For any initial values (x −i , y −i ) ∈ I 1 × I 2 for i ∈ {0, 1, 2}, the six-dimensional discrete dynamical system (5) x n+1 = f 1 (x n , x n−1 , x n−2 , y n , y n−1 , y n−2 ), y n+1 = f 2 (x n , x n−1 , x n−2 , y n , y n−1 , y n−2 ) has a unique solution {(x n , y n )} ∞ n=−2 . Definition 1. An equilibrium point of system (5) is a point (x, y) that satisfies x, x, y, y, y). Together with the system (5), if we consider the associated vector map F = (f 1 , x n , x n−1 , f 2 , y n , y n−1 ), then the point (x, y) is also called a fixed point of the vector map F . Definition 2. Let (x, y) be an equilibrium point of the map F where f 1 and f 2 are continuously differentiable functions at (x, y). The linearized system of (5) about the equilibrium point (x, y) is where X n = (x n , x n−1 , x n−2 , y n , y n−1 , y n−2 ) T and B is a Jacobian matrix of the system (5) about the equilibrium point (x, y).
Definition 3. Let X be a fixed point of the system of difference equations (6). If no eigenvalues of the Jocobian matrix B about X has absolute value equal to one, then X is called hyperbolic. If there exists an eigenvalue of the Jocobian matrix J F about X with absolute value equal to one, then X is called non-hyperbolic.
Theorem 1. Let X be a fixed point of the system of difference equations (6). If all eigenvalues of the Jocobian matrix J F about X lie inside the open unit disk |λ| < 1, then X is locally asymtotically stable. If one of them has a modulus greater than one, then X is unstable.
Theorem 2 (Routh-Hurwitz criterion). Assume that X n+1 = F (X n ), n = 0, 1, . . . , is a system of difference equations and X is a fixed point of F , the characteristic polynomial of this system about the equilibrium point X is P (λ) = a 0 λ n + a 1 λ n−1 + · · · + a n−1 λ + a n = 0, with real coefficients and a 0 > 0. Then all roots of the polynomial P (λ) lie inside the open unit disk |λ| < 1 if and only if ∆ k > 0 for k = 1, 2, . . . , n where ∆ k is the principal minor of order k of the n × n matrix For other basic knowledge about difference equations and their systems, the reader is referred to [19,20,21].

Stability Nature of Equilibrium Points
In this section, we shall state the equilibrium points of system (3) and we will investigate the stability character of these points. One can easily see that the values of the equilibrium points depends on r and s. Lemma 1. We have the following: 84 Qualitative study of a third order rational system of difference. . .
Before we investigate local asymptotic stability of the aforementioned equilibrium points, we shall build the corresponding linearized form of the system (3) and consider the following transformation The Jacobian matrix about the fixed point (x, y) under this transformation is as follows where r, s ∈ (0, ∞).
Proof. The linearized system of (3) about the equilibrium point (x 0 , y 0 ) is given by The characteristic equation associated with J F (x 0 , y 0 ) is All roots of P (λ) are equal to zero. Since all eigenvalues of the Jacobian matrix J F about (0, 0) lie inside the open unit disk |λ| < 1, the zero equilibrium point is locally asymptotically stable and this completes the proof. Proof. The linearized system of (3) about the equilibrium point (x 1 , y 1 ) = ( s+1 rs−1 , r+1 rs−1 ) is given by where X n = (x n , x n−1 , x n−2 , y n , y n−1 , y n−2 ) T and The characteristic equation of J F (x 0 , y 0 )(x 1 , y 1 ) is as follows: From this characteristic equation, we obtain It is clear that not all of ∆ 6×6 > 0. Therefore, by Theorem 2, the unique positive equilibrium point (x 1 , y 1 ) is unstable. But as the product of roots equals − 1 rs , we conclude that there is at least one root with modulus less than one. Therefore, (x 1 , y 1 ) is a Saddle point. This completes the proof.

Global stability of the zero equilibrium point
Consider the subsets I j , j ∈ {1, 2, 3, 4} in R, where Proof. We show that I 1 is an invariant subset of R 2 for system (3). The proof is by induction on n.
Let (x −i , y −i ) ∈ I 1 for i ∈ {0, 1, 2}. Then That is (x n+1 , y n+1 ) ∈ I 1 . Therefore, I 1 is an invariant subset of R 2 for system (3). Similarly, we can show that I 2 is an invariant subset of R 2 for system (3). This completes the proof.
We have shown that the zero equilibrium point is locally asymptotically stable. In the following results, we shall show that the zero equilibrium point is a global attractor with basin as well as it is globally asymptotically stable.
We put the following result without proof.  Proof. The proof is a direct consequence of Theorem 3 and Theorem 7.
In the following result, we shall show that there exist unbounded solutions of system (3).

Periodicity
In the following result, we show that under certain conditions, system (3) possesses period-2 solutions.
Theorem 10. Assume that rs > 1 in system (3). Then we have the following: (  {(p 1 , q 1 ), (p 2 , q 2 ), (p 1 , q 1 ), (p 2 , q 2 ), . . . } is a period-2 solution of system (3), where at least p 1 = p 2 or q 1 = q 2 . Then we have If p i , q i are positive real numbers for i = 1, 2, then after some calculations using (7) and (2) we can obtain p 1 = p 2 = x 1 and q 1 = q 2 = y 1 which is a contradiction of our assumption. This means that, period-2 solution of the form (7) with positive real numbers p i , q i , i = 1, 2 does not exist.

Rate of Convergence
In this section, we shall study the rate of convergence of a solution that converges to the equilibrium point (0, 0) of the system (3). The following result gives the rate of convergence of the solution of a system of difference equations: where X n is a six-dimensional vector, A ∈ C 6×6 is a constant matrix and B : Z + → C 6×6 is a matrix function satisfying where . denotes any matrix norm which is associated with the vector norm. Also . denotes the Euclidean norm in R 2 given by 90 Qualitative study of a third order rational system of difference. . . Theorem 11 ([25]). Assume that condition (10) holds, if X n is a solution of (9), then either X n = 0 for all large n or θ = lim n→∞ n X n exists and θ is equal to the modulus of one the eigenvalues of the matrix A.
Theorem 12 ([25]). Assume that condition (10) holds, if X n is a solution of (9), then either X n = 0 for all large n or θ = lim n→∞ X n+1 X n exists and θ is equal to the modulus of one the eigenvalues of the matrix A.
Assume that lim n→∞ x n = x and lim n→∞ y n = y and consider the error vector We shall find a system satisfied by the error terms. The error terms satisfy the relations where A 0n = 0, A 1n = r(x n−1 + x) 1 + y , A 2n = 0, We can write system (11) as C in e 1 n−i + That is we can write where a n → 0, b n → 0, c n → 0 and d n → 0 as n → ∞.
Then we can write The matrix K is the same as the Jacobian matrix evaluated at the zero equilibrium point. Using Theorems 11 and 12, we have the following result.
n=−2 of the system (3) satisfies both of the following asymptotic relations: where the eigenvalues of the Jacobian matrix evaluated at the equilibrium point (0, 0) are all equal to zero.

Numerical simulations
In order to verify our theoretical results we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions of the system (3). All plots in this section are drawn with Mathematica.

Conclusion and some open problems
In this paper, some properties of a higher dimensional difference system of equations were studied. Namely, we investigated the equilibria of the system (3) in details. Also, we investigated the stability character of these points using the linearization method . The main goal of dynamical systems theory is to approach the global behavior and the rate of convergence. Therefore, here we studied the global asymptotic stability and the rate of convergence of the zero equilibrium point of the system. Also, the existence of unbounded solutions and the periodicity of solutions of the system were studied. Even if it will be possible to obtain analytical results, it would be quite difficult to deal with them. So, numerical simulations were used to verify the correctness of analytical results.
Finally, we conjecture that our study can be extended to a system with higher order. We shall give some interesting open problems for difference systems of equations.
We conjecture that the obtained results are satisfied to the system where the parameters α, β are non-negative real numbers, the initial conditions x −i , y −i are non-negative real numbers, i = 0, 1, . . . , max{2l, 2k + 1} and with natural numbers l ≥ 1, k ≥ 1.