Stability and Periodic Nature of A System of Difference Equations

Kırklareli University, Pınarhisar Vocational School of Higher Education, 39300, Kırklareli, Turkey erkantasdemir@hotmail.com or erkan.tasdemir@klu.edu.tr Abstract In this paper, we investigate the equilibrium points of following a system of difference equations xn+1 = xn 2yn − 1, yn+1 = yn 2xn − 1. We also study the asymptotic stability of related system of difference equations. Further we examine the periodic solutions of related system with period two. Additionally, we find out the invariant interval and periodic cycles of related system of difference equations. 2010 AMS Classification: 39A10, 39A30, 39A23


Introduction
Difference equations and their systems play a crucial role in different fields of science. Many scientific fields need mathematical models to interpret their results. Especially mathematical models via discrete variables are related to this topic. For the last decades, many scientists have studied stability of equilibrium points, periodicity and boundedness of difference equations or their systems. There are many paper related to difference equations and their systems for examples : Xianyi et al, in [30], investigated the global asymptotic stability of following rational difference equation x n x n−1 + α x n + x n−1 .

Taşdemir
They obtained a sufficient condition for global asymptotic stability of related system. De Vault et al, in [4], investigated global behaviours of following difference equation y n+1 = A + y n y n−1 .
Abu Saris et al, in [1], studied global asymptotic stability of following difference equation Papaschinopoluos et al, in [21], considered the following system of difference equations They obtained results related to global asymptotic stability of positive equilibrium point. Kent et al, in [10], studied long-term behaviours of solutions of difference equation Moreover, in [31], Wang et al and in [18], Liu et al examined convergence of solutions of related difference equation about equilibrium points.
In [12], Kent et al investigated the periodicity of solutions, existence of bounded or unbounded of solutions and stability of solutions of difference equation Kent et al, in [13], studied the periodicity, stability and unbounded solutions of difference equation Further, there are many books and papers related to dynamical systems, see [1] - [32]. In this paper, we investigate the equilibrium points of following a system of difference equations x n+1 = x n−2 y n − 1, y n+1 = y n−2 x n − 1, n = 0, 1, ..., (1.1) where all initial values are real numbers. We also study the asymptotic stability of related system of difference equations. Furthermore, we examine the existence of periodic solutions of related system. From here to the end of this section, we show useful definitions and theorems which are used during this study.
Firstly, let us introduce discrete dynamical system of the form implying |x n −x| < ε and |y n −ȳ| < ε for n ∈ N.
and B is a Jacobian matrix of system (1.2) about the equilibrium point (x,ȳ).
is a system of difference equations such thatX is a fixed point of F .
(i) If all eigenvalues of the Jacobian matrix B aboutX lie inside the open unit disk |λ| < 1, that is, if all of them have absolute value less than one, thenX is locally asymptotically stable.
(ii) If at least one of them has a modulus greater than one, thenX is unstable.

Equilibrium Points of System (1.1)
In this here, we examine the equilibrium points of System (1.1). System (1.1) has two equilibrium points such that 618, the elements of second equilibrium point is equal to the Golden Ratio.

Existence of Periodic and Bounded Solutions of System (1.1)
In this section, we investigate the periodic behaviours of solutions of System (1.1). Firstly we find out the two periodic solutions of System (1.1). Further, we determine existence of bounded of solutions of System (1.1). Moreover we study the periodic cycles of solutions of System (1.1).

Theorem 3.1. System (1.1) has periodic solutions with period two.
Proof . Assume that system (1.1) has two periodic solutions. Thus we have for n ≥ 0: where a ̸ = b and c ̸ = d. Hence we get from system (1.1) and (3.1): Therefore we obtain from (3.2)-(3.3) a = c and similarly from (3.4)-(3.5) b = d. According to these, we can write the following equations: Now, we write (3.7) into (3.6). Thus we have the following: When this equation rearrange, we obtain that From this we obtain the four roots of (3.8) as If a = 0 or a = −1, then System (1.1) has two periodic cycle such as Since the other values of a are equal to elements of eqilibrium points, they are equilibrium solutions. So, the proof completed as desired. □ Now we investigate the invariant interval of System (1.1).
From these and by induction, we obtain that x n , y n ∈ (−1, 0) for all n ≥ 1. □

Remark 3.3.
There are two equations corresponding to the odd and even arguments of x n and y n such that Firstly we discuss the odd terms of x n . Hence we have from System (1.1): Similarly we can write the even terms of x n and the odd and even terms of y n as follows: Therefore x 2n+1 and x 2n+2 satisfy the following equation where n ∈ N and u n , v n ∈ (−1, 0). Likewise y 2n+1 and y 2n+2 satisfy the equation where n ∈ N and u n , v n ∈ (−1, 0).
From (3.13) and (3.14), we obtain the following eight equilibrium points: Now we take the f and g functions corresponding to (3.13) and (3.14) respectively: where u, v, w ∈ (−1, 0). Then we obtain the followings: So, the f and g functions are strictly increasing in each argument.
Now we will prove (a). The proofs of (b), (c) and (d) are similar to (a), so we leave it to readers. We know from Theorem 3.4: u n ∈ (ū 3 ,ū 2 ), n ≥ N. We assume that I =ū 3 . From (3.15), there is an ε > 0 such that Since f is a monotonic function, we can write Therefore we obtain by induction u n > I +ε, n ≥ N . And so lim n→∞ inf u n ≥ I +ε. It is a contradiction.
We assume that I ∈ (ū 3 ,ū 2 ). Let (u n k ) k∈N be a subsequence of (u n ) ∞ n=−1 such that lim

Stability Analysis of System (1.1)
This section, we study the stability of System (1.1). Moreover, we determine that both negative and positive equilibrium points of System (1.1) are unstable. Proof . Firstly we study linearized form of System (1.1). For this, we consider the transformation: Therefore we have the Jacobian matrix about equilibrium point (x,ȳ): is X N +1 = B(x,ȳ)X n where X n = ((x n , x n−1 , x n−2 , y n , y n−1 , y n−2 )) T and Hence, we have six roots of Eq.(4.1): Therefore we have the Jacobian matrix about equilibrium point (x,ȳ): Because of |λ 5 | < |λ 3 | = |λ 4 | < 1 < |λ 1 | = |λ 2 | < |λ 6 | , and from linearized stability theorem, three roots of the characteristic equation lie inside the unit disk but the other roots lie outside the unit disk. So, the positive equilibrium of System (1.1) is locally unstable. □

Conclusion
In this study, we determine the equilibrium points of System (1.1). We examine the periodicity of solutions of System (1.1) with period two. Moreover we find out that if the initial values of System (1.1) be in (−1, 0) then the solutions of System (1.1) have bound from above and below. The solutions of System (1.1) also converge to two periodic cycle if the initial values in (−1, 0). Finally, we investigate the stability of two equilibrium points of System (1.1).