Int. J. Anal. Appl. (2023), 21:17 On the Behavior of the Nonlinear Difference Equation yn+1 = Ayn−1 +Byn−3 + Cyn−1+Dyn−3 Fyn−3−E Turki D. Alharbi1,2,∗, Elsayed M. Elsayed1,3 1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia 2Department of Mathematics, University College in Al-Leeth, Umm Al-Qura University, Makkah, Saudi Arabia 3Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt ∗Corresponding author: tdharbi@uqu.edu.sa Abstract. The theory of difference equations got a significant position in the applicable analysis. Therefore, studying the qualitative behavior of the difference equations is a fruitful area of research that has increasingly attracted many researchers. In this paper, we demonstrate the stability and the existence of periodic solutions of the nonlinear difference equation. Moreover, we provide some numerical simulations to confirm our results. 1. Introduction The major purpose of this study is to provide a substantial analysis on periodicity of solution, local asymptotic stability and global behavior of the following difference equations yn+1 = Ayn−1 +Byn−3 + Cyn−1 +Dyn−3 Fyn−3 −E , n =0,1, ... (1.1) where the parameters A, B, C , and D are positive real numbers and the initial conditions y−3,y−2,y−1, and y0 are positive real. The study of difference equations is of utmost importance in mathematical applications. These equations also naturally appear as discrete analogs and as numerical solutions of some dynamical systems of differential equations that illustrate several phenomena in physics, biology, ecology, engi- neering, economics, etc. [1–10]. The theory of difference equations occupied a central position in Received: Jan. 18, 2023. 2020 Mathematics Subject Classification. 39A11. Key words and phrases. difference equation; recursive sequences; stability; periodicity; boundedness. https://doi.org/10.28924/2291-8639-21-2023-17 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-17 2 Int. J. Anal. Appl. (2023), 21:17 the applicable analysis. So, there is no doubt that the theory of discrete time equations will persist in playing an important role in mathematics. Therefore, it has been developing in terms of analysing the behavior and solving these equations. This progress can obviously be seen in the published studies, take for instance, Alharbi et al. [11] analysed the stability and the periodicity of solutions and explored the form of solution for a special case of the rational difference equation Zn+1 = aZn−5 − bZn−5 cZn−5 −dXn−11 , n =0,1, ... El-Dessoky [12] obtained the local and global stability of the positive solutions, the periodic behavior, and the boundedness character of the following difference equation xn+1 = βxn−l +αxn−k + axn−t bxn−t +c , n =0,1, ... Elsayed et al. [13] investigated the stability and periodicity as well as obtaining the solutions of a higher-order difference equation Un+1 = Un−9Un−5Un−1 Un−7Un−3(±1±Un−9Un−5Un−1) , n =0,1, ... In [14], Zayed et al. studied some qualitative properties of the solutions for the non-linear difference equation xn+1 = Axn +Bxn−k +Cxn−l +Dxn−σ + bxn−k +hxn−l dxn−k +exn−l , n =0,1, ... The boundedness solution, local stability, and global attractivity of the following second-order fractional equation xn+1 = α+γxn−1 Bxn +Dxnxn−1 +xn−1 , n =0,1, ... are demonstrated in [15] by Kostrov et al. Avotina [16] presented the periodic solution of three special cases of the rational difference equation: xn+1 = α+βxn +γxn−1 A+Bxn +Cxn−1 , n =0,1, ... For more recent studies, we refer the reader to [17-43] and references cited therein. 2. Preliminaries and Notation In this section, we introduce some definitions and theorems of the theory of difference equations that be utilized in our analysis. Assume that S be a continuously differentiable function such that S : [a,b]k+1 → [a,b], where [a,b] is a real numbers interval and k is a positive integer. Then the difference equation tn+1 = S(tn,tn−1, ...,tn−k), n =0,1,2, ... (2.1) has a unique solution {tn}∞n=−k for all set of initial values t−k,t−k+1, ...,t0 ∈ [a,b]. (Kocic and Ladas [5]) Int. J. Anal. Appl. (2023), 21:17 3 Definition 2.1. (Equilibrium Point) A point t∗ ∈ [a,b] is called an equilibrium point of equation (2.1) if t∗ = S(t∗,t∗, ...,t∗). That is, tn = t∗ for all n ≥ 0, is a solution of equation (2.1), or equivalently, t∗ is a fixed point of S. Definition 2.2. (Stability) The equilibrium point t∗ of equation 2.1 is said to be • Locally stable if, for every α > 0, there exists β > 0 such that for every t−k,t−k+1, ...,t−1,t0 ∈ [a,b] with |t−k − t∗|+ |t−k+1 − t∗|+ ...+ |t0 − t∗| < β, we have |tn − t∗| < α ∀n ≥−k. • Locally asymptotically stable if t∗ is locally stable solution of equation 2.1 and there exists µ > 0 such that for every t−k,t−k+1, ...,t−1,t0 ∈ [a,b] with |t−k − t∗|+ |t−k+1 − t∗|+ ...+ |t0 − t∗| < µ, we have lim n→∞ tn = t ∗. • Global attractor if, for every t−k,t−k+1, ...,t−1,t0 ∈ [a,b] we have lim n→∞ tn = t ∗. • globally asymptotically stable if t∗ is locally stable, and also a global attractor of equation 2.1 • unstable if t∗ is not locally stable of equation 2.1. Definition 2.3. (Periodicity) A sequence {tn}∞n=−k is a periodic solution with period q if tn+q = tn for all n ≥−k. Definition 2.4. (Linearised Equation) The linearized equation of the difference equation (2.1) about the equilibrium t∗ is the linear difference equation Xn+1 = k−1∑ i=0 ∂S(t∗,t∗, ...,t∗) ∂tn−i Xn−i (2.2) Now, suppose that the characteristic equation associated with (2.2) is Q(ζ)= Q0ζ k +Q1ζ k−1 + ...+Qk−1ζ +Qk =0 . (2.3) 4 Int. J. Anal. Appl. (2023), 21:17 Theorem A [8] Assume that Qi ∈ R, where i =1,2,3, ...,K and k ∈ {0,1,2,3,..}. Then k∑ i=1 |Qi| < 1 . is a sufficient condition for the asymptotic stability of the following the difference equation Xn+k +Q1Xn+k−1 + ...+QkXn =0 . Theorem B [9] Assume that h is a continuous function such that h : [α,β]s+1 → [α,β], where k is a positive integer and [α,β] is a real numbers interval.And consider the difference equation tn+1 = h(tn,tn−1, ...,tn−k), n =0,1,2, ... (2.4) Now, let h satisfies the following (1) For all 1≤ i ≤ k+1 where i is an integer , the function h(z1,z2, ...,zk+1) is weakly monotonic in zi for each z1,z2, ...,zk+1 . (2) Assume (m,M) is a solution of the the system m = h(m1,m2, ...,mk+1) , M = h(M1,M2, ...,Mk+1) . Then M=m, fer each (i =1,2, ...,k +1) we set mi =  m, if h is non-decreasing in zi M, if h is non-inceasing in zi, and Mi =  M, if h is non-decreasing in zi m, if h is non-inceasing in zi, m = M . So, there exists a unique fixed point t∗ of the equation (2.4) and any solution of (2.4) converges to t∗ 3. The Local Stability Analysis In this section, we calculate the equilibrium points of equation (1.1). Moreover, the local stability of these equilibrium points will be investigated. Theorem 3.1. The non-linear difference equation (1.1) has two equilibrium points y∗1 = 0 and y ∗ 2 = C+D F(1−A−B) + E F . Int. J. Anal. Appl. (2023), 21:17 5 Proof. Equation (1.1) can be written as y∗(1−A−B)= Cy∗ +Dy∗ Fy∗ −E or Fy2∗(1−A−B)−Ey∗(1−A−B)−Cy∗ −Dy∗ =0, then, Fy2∗(1−A−B)−y∗(E(1−A−B)+C +D)=0. So, The difference equation (1.1) has two equilibrium points y∗1 =0, y ∗ 2 = C +D F(1−A−B) + E F . Theorem 3.2. The first equilibrium point y∗1 =0 of the difference equation (1.1) is locally asymptot- ically stable if |−C −D| < E(1−A−B). Proof. Suppose that g(0,∞)3 → (0,∞) is a function defined as follows g(u,w)= Au +Bw + Cu +Dw Fw −E . (3.1) Differentiating g(u,w) with respect to u and w. We get gu = A+ C Fw −E , gw = B − (FCu +DE) (Fw −E)2 , substituting y∗1 =0 into gu, and gw. We get gu(y ∗ 1,y ∗ 1)= A− C E =−P1, gw(y∗1,y ∗ 1)= B − D E =−P2. Hence, the linearized equation of (1.1) about the equilibrium point y∗1 is Zn+1 +P1Zn−1 +P2Zn−3 =0 . (3.2) It follows by Theorem A that the fixed point ,y∗1, of equation (1.1) is locally asymptotically stable if |P1|+ |P2| < 1 . So, |A− C E |+ |B − D E | < 1, this implies, |AE −C +BE −D| < E. Thus, the first equilibrium point y∗1 is locally asymptotically stable if |−C −D| < E(1−A−B). The proof is completed. 6 Int. J. Anal. Appl. (2023), 21:17 Theorem 3.3. Suppose that |Cα− (C +Eα)α| < C +D−A−B. Where α =(1−A−B), then the second equilibrium point y∗2 of equation (1.1) is locally asymptotically stable. Proof. Substituting y∗2 = C+D Fα + E F into gu, and gw. We get gu(y ∗ 2,y ∗ 2)= A+ Cα C +D =−Q1, gw(y ∗ 2,y ∗ 2)= B − (C +Eα)α C +D =−Q2. Where α =(1−A−B). So, the linearized equation of (1.1) about the equilibrium point y∗2 is Zn+1 +Q1Zn−1 +Q2Zn−3 =0 . (3.3) It can be shown by Theorem A that the fixed point y∗2 of the difference equation (1.1) is locally asymptotically stable if |Q1|+ |Q2| < 1 . So, |A+ Cα C +D |+ |B − (C +Eα)α C +D | < 1, thus, |A+Cα+B − (C +Eα)α| < C +D. Therefore, the second equilibrium y∗2 is locally asymptotically stable if |Cα− (C +Eα)α| < C +D−A−B. The proof is completed. 4. Global Behaviour Analysis We dedicate this section to showing the case under which the equilibrium points y∗ of equation (1.1) are asymptotically globally stable. Theorem 4.1. The equilibrium points y∗ of the difference equation (1.1) is globally asymptotically stable if i AE +BF +D > C +BE +E ii E +C +D > F Int. J. Anal. Appl. (2023), 21:17 7 Proof. Suppose that k and r be real numbers and assume g(k,r)2 → (k,r) is a function that defined by g(u,w)= Au +Bw + Cu +Dw Fw −E . (4.1) Now, we consider two cases. Case i. Suppose that g(u,w) is increasing in u and w. Then, assume (ζ,ρ) is a solution of the following system ζ = g(ζ,ζ) , ρ = g(ρ,ρ) . So, ζ = Aζ +Bζ + Cζ +Dζ Fζ −E , ρ = Aρ+Bρ+ Cρ+Dρ Fρ−E , this gives, Fζ2(1−A−B)−Eζ(1−A−B)= ζ(C +D), (4.2) Fρ2(1−A−B)−Eρ(1−A−B)= ρ(C +D), (4.3) after subtracting (4.3) from (4.2). We get (ζ2 −ρ2)F(1−A−B)− (ζ −ρ)E(1−A−B)− (ζ −ρ)(C +D)=0, (4.4) this implies, (ζ −ρ){(ζ +ρ)F(1−A−B)−E(1−A−B)− (C +D)}=0. (4.5) Thus, when E +C +D > F, ζ = ρ . It follows by Theorem B that y∗ is globally asymptotically stable. The proof is completed. Case ii. Suppose that g(u,w) is increasing in u and it is decreasing in w. Then, assume (ζ,ρ) is a solution of the following system ζ = g(ζ,ρ) , ρ = g(ρ,ζ) . So, ζ = Aζ +Bρ+ Cζ +Dρ Fρ−E , ρ = Aρ+Bζ + Cρ+Dζ Fζ −E , this implies, ζ(1−A)(Fρ−E)−Bρ(Fρ−E)−Cζ −Dρ =0, (4.6) 8 Int. J. Anal. Appl. (2023), 21:17 ρ(1−A)(Fζ −E)−Bζ(Fζ −E)−Cρ−Dζ =0. (4.7) Now, subtracting (4.7) from (4.6). We get (ζ −ρ){AE −E +BF(ζ +ρ)−BE −C +D}=0. (4.8) Therefore, when AE +BF +D > C +BE +E ζ = ρ . It can be shown by Theorem B that y∗ is globally asymptotically stable. The proof is completed. 5. Existence of Periodic Solutions This section discusses the existence of periodic behavior of the nonlinear difference equation (1.1). The following theorem states the necessary and sufficient conditions that assure Eq.(1.1) has periodic behavior of prime period two. Theorem 5.1. The difference equation (1.1) has solution of period two if and only if E(1−A−B)+C +D 6=0 (5.1) Proof. Assume that equation (1.1) has a solution of period two ...,α,β,α,β,... with α 6= β α = Aα+Bα+ Cα+Dα Fα−E , β = Aβ +Bβ + Cβ +Dβ Fβ −E . So, Fα2(1−A−B)−Eα(1−A−B)= α(C +D) , (5.2) Fβ2(1−A−B)−Eβ(1−A−B)= β(C +D) . (5.3) Subtracting (5.3) from (5.2) gives F(1−A−B)(α2 −β2)−E(1−A−B)(α−β)= (C +D)(α−β) , this implies, F(1−A−B)(α+β)−Eα(1−A−B)= (C +D) . Consequently, α+β = E(1−A−B)+C +D F(1−A−B) . (5.4) Int. J. Anal. Appl. (2023), 21:17 9 Again, adding (5.2) and (5.3). We get F(1−A−B)(α2 +β2)= {E(1−A−B)+(C +D)}(α+β) . (5.5) By using (5.4), (5.5) , and the relation (α+β)2 = α2 +2αβ +β2, we obtain F(1−A−B){(α+β)2 −2αβ}= {E(1−A−B)+(C +D)}(α+β) , then, 2F(1−A−B)αβ = F(1−A−B)(α+β)2 −{E(1−A−B)+(C +D)}(α+β) , 2F(1−A−B)αβ = (E(1−A−B)+C +D)2 F(1−A−B) −{E(1−A−B)+C+D}( E(1−A−B)+C +D F(1−A−B) ) . Thus, αβ =0. (5.6) Therefore, it follows from equations (20) and (22) that α and β are the two distinct roots of the quadratic equation X2 − (α+β)X +αβ =0. (5.7) That is, X2 − ( E(1−A−B)+C +D F(1−A−B) )X =0, then, F(1−A−B)X2 − (E(1−A−B)+C +D)X =0, so, (E(1−A−B)+C +D)2 > 0. For (E(1−A−B)+C +D) 6=0, the condition (5.1) holds. On the other side, suppose that condition (5.1) is true. We will demonstrate that equation (1.1) has a prime period two solution. Set y−3 = y−1 = p = E(1−A−B)+C +D F(1−A−B) and y−2 = y0 = q =0. Now, we want to show that y1 = p, and y2 =0. It follows from equation (1.1) that y1 = Ap+Bp+ Cp+Dp Fp−E , so, y1 =(A+B) (E(1−A−B)+C +D F(1−A−B) ) + (C +D)( E(1−A−B)+C+D F(1−A−B) ) F( E(1−A−B)+C+D F(1−A−B) )−E , 10 Int. J. Anal. Appl. (2023), 21:17 =(A+B) (E(1−A−B)+C +D F(1−A−B) ) + (C +D)(E(1−A−B)+C +D F(C +D) , =(A+B) (E(1−A−B)+C +D F(1−A−B) ) + (E(1−A−B)+C +D F , = (E(1−A−B)+C +D F )( 1+ A+B (1−A−B) ) = E(1−A−B)+C +D F(1−A−B) = p, y2 = Aq +Bq + Cq +Dq Fq −E =0= q. So, by induction we get y2n = q and y2n+1 = p for all n ≥−3. Hence, equation (1.1) has the prime period two solution p and q. Where p and q are the distinct roots of the quadratic equation (5.7). 6. Numerical Examples In this part, we provide some examples that verify our analytical results. MATLAB programming is used to show numerically the behavior of the nonlinear difference equation(1.1). Example 6.1. Figure 1 shows the behavior of Eq.(1.1) tends to the first equilibrium point y∗1 = 0 when the parameters and the initial values are A = 0.1, B = 0.2, C = 1, D = 2, F = 4, E = 6, y−3 =−3, y−2 =2, y−1 =−0.5, and y0 =1. Example 6.2. Figure 2 presents the behavior of Eq.(1) approaches to the second equilibrium point Figure 1. The Behaviour of Equation (1.1) y∗2 =0 when we assume the parameters and the initial values that A =0.6, B =0.2, C =3, D =4, F =6, E =5, y−3 =5, y−2 =−3, y−1 =1, and y0 =4. Int. J. Anal. Appl. (2023), 21:17 11 Figure 2. The Behaviour of Equation (1.1) Example 6.3. The unstable behavior of Eq.(1.1) is shown in figure 3. we assume the parameters and the initial values that A = 0.1, B = 0.2, C = 5, D = 8, F = 0.4, E = 6, y−3 = 1, y−2 = −6, y−1 =3, and y0 =−4. Figure 3. The Behaviour of Equation (1.1) Example 6.4. In figure 4, The global stability behavior of Eq.(1.1) is shown. It is clear that the behavior of Eq.(1.1) tends to the fixed point y∗1 as n goes to ∞ under the following the initial conditions and the parameters A =0.1, B =0.2, C =1, D =2, F =4, E =8, y−3 =1, y−2 =−6, y−1 =3, and y0 =−4. 12 Int. J. Anal. Appl. (2023), 21:17 Figure 4. The Behaviour of Equation (1.1) Example 6.5. Figure 5 demonstrates the global stability behavior of the fixed point y∗2 when the initial conditions and the parameters are A = 0.2, B = 0.16, C = 0.123, D = 14, F = 0.5, E = 5, y−3 =5, y−2 =−3, y−1 =1, and y0 =−4. Figure 5. The Behaviour of Equation (1.1) Example 6.6. Figure 6 shows that Eq.(1.1) has a prime period two solution when the initial conditions and the parameters are A = 0.2, B = 0.1, C = 0.2, D = 3, F = 1, E = 0.6, y−3 = p, y−2 = q, y−1 = p, and y0 = q where p and q satisfied Theorem 5.1 Int. J. Anal. Appl. (2023), 21:17 13 Figure 6. The Behaviour of Equation (1.1) 7. Conclusion This study discusses the dynamics of the nonlinear difference equation (1.1). In section 3 we illustrated that when the local stability condition in Theorem 3.2 is satisfied, the behavior tends to the stability state of the equilibrium point y∗1 =0. While, the equilibrium y ∗ 2 will be locally asymptotically stable when |Cα − (C + Eα)α| < C + D − A − B. The global solution of the equilibrium points conditions is shown in section 4. Section 5 discussed the necessary and sufficient conditions to obtain the periodic solution of equation (1). For confirmation of our theoretical analysis, we presented some numerical examples in section 6, and figures 1-6 verified the results. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] R.E. Mickens, Difference Equations: Theory and Applications, 2nd Ed, Chapman and Hall, New York, (1990). [2] H.F. Huo, W.T. Li, Permanence and Global Stability of Positive Solutions of a Nonautonomous Discrete Ratio- Dependent Predator-Prey Model, Discr. Dyn. Nat. Soc. 2005 (2005), 135–144. https://doi.org/10.1155/ddns. 2005.135. [3] G. Ladas, G. Tzanetopoulos, A. Tovbis, On May’s Host Parasitoid Model, J. Differ. Equ. Appl. 2 (1996), 195–204. https://doi.org/10.1080/10236199608808054. [4] S. Stevic, A Global Convergence Results With Applications to Periodic Solutions, Indian J. Pure Appl. Math. 33 (2002), 45-53. [5] V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Springer Netherlands, Dordrecht, 1993. https://doi.org/10.1007/978-94-017-1703-8. [6] H. Sedaghat, Nonlinear Difference Equations, Springer Netherlands, Dordrecht, 2003. https://doi.org/10.1007/ 978-94-017-0417-5. [7] E.C. Pielou, Population and Community Ecology, Gordon and Breach, New York, (1974). https://doi.org/10.1155/ddns.2005.135 https://doi.org/10.1155/ddns.2005.135 https://doi.org/10.1080/10236199608808054 https://doi.org/10.1007/978-94-017-1703-8 https://doi.org/10.1007/978-94-017-0417-5 https://doi.org/10.1007/978-94-017-0417-5 14 Int. J. Anal. Appl. (2023), 21:17 [8] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/ CRC Press, New York, (2001). [9] EA. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations. 1st Ed, Chapman & Hall/ CRC Press, New York, (2004). [10] P. Cull, M.E. Flahive, R.O. Robson, Difference Equations: From Rabbits to Chaos, Springer, New York, (2005). [11] T.D. Alharbi, E.M. Elsayed, Forms of Solution and Qualitative Behavior of Twelfth-Order Rational Difference Equation, Int. J. Differ. Equ. 17 (2022), 281-292. [12] M.M. El-Dessoky, Studies on the Higher Order Difference Equation xn+1 = βxn−l + αxn−k + axn−t bxn−t+c , J. Comput. Anal. Appl. 29 (2021), 116-131. [13] E.M. Elsayed, B.S. Alofi, A.Q. Khan, Qualitative Behavior of Solutions of Tenth-Order Recursive Sequence Equa- tion, Math. Probl. Eng. 2022 (2022), 5242325. https://doi.org/10.1155/2022/5242325. [14] M.A. El-Moneam, E.M.E. Zayed, Dynamics of the Rational Difference Equation, Inform. Sci. Lett. 3 (2014), 45–53. https://doi.org/10.12785/isl/030202. [15] Y. Kostrov, Z. Kudlak, On a Second-Order Rational Difference Equation with a Quadratic Term, Int. J. Differ. Equ. 11 (2016), 179-202. [16] M. Avotina, On Three Second-Order Rational Difference Equations with Period-Two Solutions, Int. J. Differ. Equ. 9 (2014), 23-35. [17] A. Asiri, M.M. El-Dessoky, E.M. Elsayed, Solution of a Third Order Fractional System of Difference Equations, J. Comput. Anal. Appl., 24 (2018), 444-453. [18] S. Moranjkic, Z. Nurkanovic, Local and Global Dynamics of Certain Second-Order Rational Difference Equations Containing Quadratic Terms, Adv. Dyn. Syst. Appl. 12 (2017), 123-157. [19] M.N. Phong, A Note on a System of Two Nonlinear Difference Equations, Electron. J. Math. Anal. Appl. 3 (2015), 170-179. [20] W. Wang, J. Tian, Difference Equations Involving Causal Operators With Nonlinear Boundary Conditions, J. Non- linear Sci. Appl. 8 (2015), 267-274. [21] H.S. Alayachi, M.S.M. Noorani, A.Q. Khan, M.B. Almatrafi, Analytic Solutions and Stability of Sixth Order Differ- ence Equations, Math. Probl. Eng. 2020 (2020), 1230979. https://doi.org/10.1155/2020/1230979. [22] A.M. Alotaibi, M.A. El-Moneam, On the Dynamics of the Nonlinear Rational Difference Equation xn+1 = axn−m+δxn β+γxn−kxn−1(xn−k+xn−1) , AIMS Math. 7 (2022), 7374-7384. [23] J. Bektesevic, M. Mehuljic, V. Hadziabdic, Global Asymptotic Behavior of Some Quadratic Rational Second-Order Difference Equations, Int. J. Differ. Equ. 20 (2017), 169-183. [24] E. M. Elsayed, K. N. Alshabi and F. Alzahrani, Qualitative Study of Solution of Some Higher Order Difference Equations, J. Comput. Anal. Appl. 26 (2019), 1179-1191. [25] E.M. Elsayed, K.N. Alharbi, The Expressions and Behavior of Solutions for Nonlinear Systems of Rational Difference Equations, J. Innov. Appl. Math. Comput. Sci. 2 (2022), 78–91. [26] E.M. Elsayed, A. Alshareef, Qualitative Behavior of A System of Second Order Difference Equations, Eur. J. Math. Appl. 1 (2021), 15. https://doi.org/10.28919/ejma.2021.1.15. [27] E.M. Elsayed, N.H. Alotaibi, The Form of the Solutions and Behavior of Some Systems of Nonlinear Difference Equations, Dyn. Contin. Discr. Impuls. Syst. Ser. A: Math. Anal. 27 (2020), 283-297. [28] E.M. Elsayed, H.S. Gafel, Some Systems of Three Nonlinear Difference Equations, J. Comput. Anal. Appl. 29 (2021), 86-108. [29] E.M. Elsayed, J.G. Al-Juaid, H. Malaikah, On the Dynamical Behaviors of a Quadratic Difference Equation of Order Three, Eur. J. Math. Appl. 3 (2023), 1. https://doi.org/10.28919/ejma.2023.3.1. https://doi.org/10.1155/2022/5242325 https://doi.org/10.12785/isl/030202 https://doi.org/10.1155/2020/1230979 https://doi.org/10.28919/ejma.2021.1.15 https://doi.org/10.28919/ejma.2023.3.1 Int. J. Anal. Appl. (2023), 21:17 15 [30] E.M. Elsayed, J.G. AL-Juaid, The Form of Solutions and Periodic Nature for Some System of Difference Equations, Fundam. J. Math. Appl. 6 (2023), 24-34. https://doi.org/10.33401/fujma.1166022. [31] E.M. Elsayed, M. M. Alzubaidi, On a Higher-Order Systems of Difference Equations, Pure Appl. Anal. 2023 (2023), 2. [32] E.M. Elsayed, B. Alofi, Stability Analysis and Periodictly Properties of a Class of Rational Difference Equations, MANAS J. Eng. 10 (2022), 203-210. https://doi.org/10.51354/mjen.1027797. [33] E.M. Elasyed, M.T. Alharthi, The Form of the Solutions of Fourth Order Rational Systems of Difference Equations, Ann. Commun. Math. 5 (2022), 161-180. [34] E.M. Elsayed, A. Alghamdi. Dynamics and Global Stability of Higher Order Nonlinear Difference Equation, J. Comput. Anal. Appl. 21 (2016), 493-503. [35] E.M. Elsayed, A. Alshareef, Qualitative Behavior of A System of Second Order Difference Equations, Eur. J. Math. Appl. 1 (2021), 15. https://doi.org/10.28919/ejma.2021.1.15. [36] M. Garic-Demirovic, M. Nurkanovic, Z. Nurkanovic, Stability, Periodicity and Neimark-Sacker Bifurcation of Certain Homogeneous Fractional Difference Equations, Int. J. Differ. Equ. 12 (2017), 27-53. [37] M. Gümüş, R. Abo-Zeid, Qualitative Study of a Third Order Rational System of Difference Equations, Math. Morav. 25 (2021), 81-97. [38] S. Kalabusic, M. Nurkanovic, Z. Nurkanovic, Global Dynamics of Certain Mix Monotone Difference Equation, Mathematics, 6 (2018), 10. https://doi.org/10.3390/math6010010. [39] A. Khaliq, E. Elsayed, The Dynamics and Solution of Some Difference Equations, J. Nonlinear Sci. Appl. 9 (2016), 1052-1063. [40] W.X. Ma, Global Behavior of a Higher-Order Nonlinear Difference Equation with Many Arbitrary Multivariate Functions, East Asian J. Appl. Math. 9 (2019), 643–650. https://doi.org/10.4208/eajam.140219.070519. [41] S. Moranjkic, Z. Nurkanovic, Local and Global Dynamics of Certain Second-Order Rational Difference Equations Containing Quadratic Terms, Adv. Dyn. Syst. Appl. 12 (2017), 123-157. [42] M. Saleh, A. Farhat, Global Asymptotic Stability of The Higher Order Equation xn+1 = axn+bxn−k A+Bxn−k , J. Appl. Math. Comput. 55 (2017), 135-148. [43] E.M.E. Zayed, On the Dynamics of a New Nonlinear Rational Difference, Dyn. Contin. Discr. Impuls. Syst. Ser. A. Math. Anal., 27 (2020), 153-165. https://doi.org/10.33401/fujma.1166022 https://doi.org/10.51354/mjen.1027797 https://doi.org/10.28919/ejma.2021.1.15 https://doi.org/10.3390/math6010010 https://doi.org/10.4208/eajam.140219.070519 1. Introduction 2. Preliminaries and Notation 3. The Local Stability Analysis 4. Global Behaviour Analysis 5. Existence of Periodic Solutions 6. Numerical Examples 7. Conclusion References