Global Character of a Six-Dimensional Nonlinear System of Difference Equations

The aim of this paper is to study the dynamical behavior of positive solutions for a system of rational difference equations of the following form: , , , where the parameters and the initial values for are positive real numbers.


Introduction
Higher-order fractional difference equations are of great importance in applications.Such equations also seem surely as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering, economics, and so on.For example, see [1,2].The theory of difference equations is the focus point in applicable analysis.That is, the importance of the theory of difference equations in mathematics as a whole will go on.Hence, it is very valuable to investigate the behavior of solutions of a system of fractional difference equations and to present the stability character of equilibrium points.
In the recent times, the behavior of solutions of various systems of rational difference equations has been one of the main topics in the theory of difference equations (see [3][4][5][6][7] and the references cited therein).
According to us, it is of great importance to investigate not only rational nonlinear difference equations and their systems, but also those equations and systems which contain powers of arbitrary positive numbers (see [8][9][10][11][12]).
In [13], Kurbanlı et al. studied the behavior of positive solutions of the following system of difference equations: where the initial conditions are arbitrary nonnegative real numbers.
In [15], Zhang et al. studied the dynamical behavior of positive solutions for a system for third-order rational difference equations: In [16], Din et al. studied the dynamics of a system of fourth-order rational difference equations: In [17], Touafek and Elsayed investigated the behavior of solutions of systems of difference equations: with a nonzero real number's initial conditions.In [18], El-Owaidy et al. investigated the global behavior of the following difference equation: with nonnegative parameters and nonnegative initial values.Motivated by all above-mentioned works, in the paper we investigate the equilibrium points, the local asymptotic stability of these points, the global behavior of positive solutions, the existence unbounded solutions, and the existence of the prime two-periodic solutions of the following system: where the parameters , , ,  1 ,  1 ,  1 ,  and the initial values  − , V − for  = 0, 1, 2 are positive real numbers.Our results extend and complement some results in the literature.
By using the induction and the equations in (8) we see that if  − ,  − are positive real numbers for  ∈ {0, 1, 2}, then which means that positive initial values generate positive solutions of system (8).
As far as we examine, there is no paper dealing with system (7).Therefore, in this paper we focus on system (7) in order to fill in the gap.

Preliminaries
For the completeness in the paper, we find it useful to remember some basic concepts of the difference equations theory as follows.
(i) An equilibrium point (, ) is said to be stable if, for every  > 0, there exists  > 0 such that for every initial value where and  is a Jacobian matrix of system (10) about the equilibrium point (, ).
Definition 4. For the system  +1 = (  ),  = 0, 1, . .., of difference equations such that  is a fixed point of , if no eigenvalues of the Jacobian matrix  about  have absolute value equal to one, then  is called hyperbolic.If there exists an eigenvalue of the Jacobian matrix  about  with absolute value equal to one, then  is called nonhyperbolic.
The following result, known as the Linearized Stability Theorem, is very practical in confirming the local stability character of the equilibrium point (, ) of system (10).
Theorem 5.For the system  +1 = (  ),  = 0, 1, . .., of difference equations such that  is a fixed point of , if all eigenvalues of the Jacobian matrix  about  lie inside the open unit disk || < 1, then  is locally asymptotically stable.If one of them has a modulus greater than one, then  is unstable.
For other basic knowledge about difference equations and their systems, the reader is referred to books [19][20][21][22].

Main Results
In this section we prove our main results.Theorem 6.One has the following cases for the equilibrium points of ( 8): (i) ( 0 ,  0 ) = (0, 0) is always the equilibrium point of system (8).
Proof.The proof is easily obtained from the definition of equilibrium point.
Before we give the following theorems about the local asymptotic stability of the aforementioned equilibrium points, we build the corresponding linearized form of system (8) and consider the following transformation: where ,  1 =   , and  2 =  −1 .The Jacobian matrix about the fixed point (, ) under the above transformation is as follows: where , ,  ∈ (0, ∞).
Proof.(i) The linearized system of (8) about the equilibrium point ( 0 ,  0 ) is given by where The characteristic equation of ( 0 ,  0 ) is as follows: The roots of () are Since all eigenvalues of the Jacobian matrix  about ( 0 ,  0 ) lie inside the open unit disk || < 1, the zero equilibrium point is locally asymptotically stable.
(ii) It is easy to see that if  > 1 or  > 1, then there exists at least one root  of () such that || > 1. Hence by Theorem 5, if  > 1 or  > 1, then ( 0 ,  0 ) is unstable.Thus, the proof is complete.
(ii) The proof is similar to the proof of (i), so it will be omitted.
(ii) The proof is similar to the proof of (i), so it will be omitted.
(iii) The proof is easily seen from the proof of (i).(iv) The proof is easily seen from the proof of (i).Now, we will study the global behavior of zero equilibrium point.
Theorem 10.If  < 1 and  < 1, then the zero equilibrium point is globally asymptotically stable.
This completes the proof.
Definition 3. Let (, ) be an equilibrium point of the map  where  1 and  2 are continuously differentiable functions at (, ).The linearized system of (10) about the equilibrium point (, ) is