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In this paper, we considered some systems of rational difference equations of higher order as follows$ \begin{eqnarray*} u_{n+1} & = &\frac{v_{n-6}}{1\pm v_{n}u_{n-1}v_{n-2}u_{n-3}v_{n-4}u_{n-5}v_{n-6}}, \\ v_{n+1} & = &\frac{u_{n-6}}{1\pm u_{n}v_{n-1}u_{n-2}v_{n-3}u_{n-4}v_{n-5}u_{n-6}}, \end{eqnarray*} $where the initial conditions $ u_{0, } $ $ u_{-1}, $ $ u_{-2}, $ $ u_{-3}, $ $ u_{-4}, $ $ u_{-5}, $ $ u_{-6}, $ $ v_{0, } $ $ v_{-1}, $ $ v_{-2}, $ $ v_{-3}, $ $ v_{-4}, $ $ v_{-5} $ and $ v_{-6} $ were arbitrary real numbers. We obtained a closed form of the solutions for each considered system and also some periodic solutions of some systems were found. We presented some numerical examples to explain the obtained theoretical results.

In this paper, we considered some systems of rational difference equations of higher order as follows u n + 1 = v n − 6 1 ± v n u n − 1 v n − 2 u n − 3 v n − 4 u n − 5 v n − 6 , v n + 1 = u n − 6 1 ± u n v n − 1 u n − 2 v n − 3 u n − 4 v n − 5 u n − 6 , where the initial conditions u 0 , u − 1 , u − 2 , u − 3 , u − 4 , u − 5 , u − 6 , v 0 , v − 1 , v − 2 , v − 3 , v − 4 , v − 5 and v − 6 were arbitrary real numbers. We obtained a closed form of the solutions for each considered system and also some periodic solutions of some systems were found. We presented some numerical examples to explain the obtained theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss a nonlinear rational difference equation system with three variables. Firstly, by utilizing Jacobian theory, the stability of the system is obtained. Then, we explore the saddle point, nonhyperbolic point, and prime period‐two solution and its boundedness. Finally, we verify the results obtained in the paper via computer numerical calculations. [ABSTRACT FROM AUTHOR]

In this paper we show that the following four-dimensional system of difference equations x n + 1 = y n α z n − 1 β , y n + 1 = z n γ t n − 1 δ , z n + 1 = t n ϵ x n − 1 μ , t n + 1 = x n ξ y n − 1 ρ , n ∈ N 0 , where the parameters α, β, γ, δ, ϵ, μ, ξ, ρ ∈ ℤ and the initial values x–i, y–i, z–i, t–i, i ∈ {0, 1}, are real numbers, can be solved in closed forms, extending further some results in literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper. [ABSTRACT FROM AUTHOR]

We propose some theoretical explanations pertaining to the representation for the solution of the system of the higher-order difference equations x n + 1 = 1 A − y n − k , y n + 1 = 1 A − x n − k , n , k ∈ N 0 , where ℕ0 = ℕ ∪ {0}, A ∈ ℕ1 and the initial conditions x–k, x–k+1, ..., x0, y–k, y–k+1, ..., y0 are non zero real numbers such that their solution is related to a generalized Balancing numbers. We also study the stability character and asymptotic behavior of this system. [ABSTRACT FROM AUTHOR]

This study is involved with a class of three-dimensional system of difference equations incorporating quadratic term, which naturally extends and improve several results in the literature. Firstly, we demonstrate the existence of fixed points, the boundedness, persistence and invariance of positive solution of the mentioned system. Later, for this system, we give the global asymptotic stability at fixed point and the rate of convergence result which play an important role in the discrete dynamical systems. And lastly, some numerical examples are given to validate the effectiveness and feasibility of the theoretical findings. [ABSTRACT FROM AUTHOR]

We present six classes of nonlinear two-dimensional systems of difference equations which are solvable. Some methods for finding their general solutions are described in detail. [ABSTRACT FROM AUTHOR]

In this paper, we study the global behavior of positive solutions of the following system of difference equations ..., where the parameter A>0, the initial values x-i,y-i, z-i, i ∈ {0,1,2, . . ., k}, are arbitrary positive real number and k ∈ N. Moreover, we provide semi-cycle analysis of positive solutions to the above system of difference equations. Finally, we also give some numerical examples which support our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we derive the well-defined solutions to a θ -dimensional system of difference equations. We show that, the well-defined solutions to that system are represented in terms of Fibonacci and Lucas sequences. Moreover, we study the global stability of the solutions to that system. Finally, we give some numerical examples which confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on obtaining the closed-form solution for the following bidimensional system of higher-order rational difference equations: u(1) n+1 = 1 3 - 2u(2) n-m, u(2) n+1 = 1 3 - 2u(1) n-m, n,m N0, where the initial values u(1) -j and u(2) -j, j {0, 1, ...,m} are real numbers not equal to 3/2. We show that the solutions of this system are associated with Mersenne numbers and/or Mersenne-Lucas numbers. It is shown that the global stability of positive solutions of this system holds. Finally, we provide numerical examples to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper investigates both the local and global stability of a system of rational difference equations and its connection to co-balancing numbers. The study delves into the intricate dynamics of mathematical models and their stability properties, emphasizing the broader implications of global stability. Additionally, the investigation extends to the role of co-balancing numbers, elucidating their significance in achieving equilibrium within the solutions of the rational difference equations. The interplay between global stability and co-balancing numbers forms a foundational aspect of the analysis. The findings contribute to a deeper understanding of the mathematical structures underlying dynamic systems and offer insights into the factors influencing their stability and equilibrium. This article serves as a valuable resource for mathematicians, researchers, and scholars interested in the intersection of global stability and co-balancing sequences in the realm of rational difference equations. Moreover, the presented examples and figures consistently demonstrate the global asymptotic stability of the equilibrium point throughout the paper.

We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a specific case of systems involving Riccati difference equations of order $ m+1 $, as discussed in the literature. However, our focus was on coefficients dependent on the Jacobsthal sequence. Importantly, this system of difference equations represents a natural extension of the corresponding one-dimensional difference equation, uniquely characterized by its theoretical solvability in a closed form. Our primary objective was to demonstrate a direct linkage between the solutions of this system and Jacobsthal and Lucas-Jacobsthal numbers. The system's capacity for theoretical solvability in a closed form enhances its distinctiveness and potential applications. To accomplish this, we detailed offer theoretical explanations and proofs, establishing the relationship between the solutions and the Jacobsthal sequence. Subsequently, our exploration addressed key aspects of the Jacobsthal system, placing particular emphasis on the local stability of positive solutions. Additionally, we employed mathematical software to validate the theoretical results of this novel system in our research.

In this study, we investigate the boundedness, persistence of positive solutions, local and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the following difference equations systems of exponential forms: $ \begin{equation*} \Upsilon_{n+1} = \frac{\Gamma_{1}+\delta_{1}e^{-\Psi_{n-1}}}{\Theta_{1}+\Psi_{n}}, \ \Psi_{n+1} = \frac{\Gamma_{2}+\delta_{2}e^{-\Omega_{n-1}}}{\Theta_{2}+\Omega_{n}}, \ \Omega_{n+1} = \frac{\Gamma_{3}+\delta_{3}e^{-\Upsilon_{n-1}}}{\Theta_{3}+\Upsilon_{n}}, \end{equation*} $ $ \begin{equation*} \Upsilon_{n+1} = \frac{\Gamma_{1}+\delta_{1}e^{-\Psi_{n-1}}}{\Theta_{1}+\Upsilon_{n}}, \ \Psi_{n+1} = \frac{\Gamma_{2}+\delta_{2}e^{-\Omega_{n-1}}}{\Theta_{2}+\Psi_{n}}, \ \Omega_{n+1} = \frac{\Gamma_{3}+\delta_{3}e^{-\Upsilon_{n-1}}}{\Theta_{3}+\Omega_{n}}, \end{equation*} $ for $ n\in \mathbb{N}_{0} $, where the initial conditions $ \Upsilon_{-j} $, $ \Psi_{-j} $, $ \Omega_{-j} $, for $ j\in\{0, 1\} $ and the parameters $ \Gamma_{i} $, $ \delta_{i} $, $ \Theta_{i} $ for $ i\in\{1, 2, 3\} $ are positive constants.

This paper investigates both the local and global stability of a system of rational difference equations and its connection to co-balancing numbers. The study delves into the intricate dynamics of mathematical models and their stability properties, emphasizing the broader implications of global stability. Additionally, the investigation extends to the role of co-balancing numbers, elucidating their significance in achieving equilibrium within the solutions of the rational difference equations. The interplay between global stability and co-balancing numbers forms a foundational aspect of the analysis. The findings contribute to a deeper understanding of the mathematical structures underlying dynamic systems and offer insights into the factors influencing their stability and equilibrium. This article serves as a valuable resource for mathematicians, researchers, and scholars interested in the intersection of global stability and co-balancing sequences in the realm of rational difference equations. Moreover, the presented examples and figures consistently demonstrate the global asymptotic stability of the equilibrium point throughout the paper. [ABSTRACT FROM AUTHOR]

We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a specific case of systems involving Riccati difference equations of order m + 1, as discussed in the literature. However, our focus was on coefficients dependent on the Jacobsthal sequence. Importantly, this system of difference equations represents a natural extension of the corresponding one-dimensional difference equation, uniquely characterized by its theoretical solvability in a closed form. Our primary objective was to demonstrate a direct linkage between the solutions of this system and Jacobsthal and Lucas-Jacobsthal numbers. The system's capacity for theoretical solvability in a closed form enhances its distinctiveness and potential applications. To accomplish this, we detailed offer theoretical explanations and proofs, establishing the relationship between the solutions and the Jacobsthal sequence. Subsequently, our exploration addressed key aspects of the Jacobsthal system, placing particular emphasis on the local stability of positive solutions. Additionally, we employed mathematical software to validate the theoretical results of this novel system in our research. [ABSTRACT FROM AUTHOR]

In this paper, we represent the admissible solutions of the system of second-order rational difference equations given below in terms of Lucas and Fibonacci sequences: ...xn+1 = Lm+2+Lm+1yn􀀀1 Lm+3+Lm+2yn􀀀1; yn+1 = Lm+2+Lm+1zn􀀀1 Lm+3+Lm+2zn􀀀1; zn+1 = Lm+2+Lm+1wn􀀀1 Lm+3+Lm+2wn􀀀1; wn+1 = Lm+2 +Lm+1xn􀀀1 Lm+3 +Lm+2xn􀀀1: where n 2 N0, fLmg+¥ m=􀀀¥ is Lucas sequence and the initial conditions x􀀀1, x0, y􀀀1, y0, z􀀀1, z0, w􀀀1, w0 are arbitrary real numbers such that v􀀀i 6= 􀀀 Lm+3 Lm+2, where v􀀀i = x􀀀i;y􀀀i; z􀀀i;w􀀀i, i = 0;1 and m 2 Z. [ABSTRACT FROM AUTHOR]

In this paper, we construct and formulate the solutions and periodicity character of the following nonlinear rational systems of difference equations: $ \begin{equation} S_{n+1} = \dfrac{T_{n} S_{n-2}}{S_{n-2} + T_{n-1}},\quad T_{n+1} = \dfrac{S_{n} T_{n-2}}{\pm T_{n-2} \pm S_{n-1}},\quad \;\;\;\; n = 0,1,2,...., \quad\quad\quad(0.1)\end{equation} $ where the initial conditions $ s_{-2}, s_{-1}, s_0, t_{-2}, t_{-1}, t_0 $ are positive real numbers. Moreover, some mathematical programs are used to support our theoretical results of each system in this paper.

In this paper, we study the following two-dimesional system of difference equations ... where the parameters a;b; c;d and the initial values x i; y i, i 2 f1;2;3;4;5;6g, are real numbers. We show that some subclasses of nonlinear two-dimensional system of difference equations are solvable in closed form. We also describe the forbidden set of solutions of the system of differ- ence equations. Some numerical examples are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

We provide some theoretical justifications pertaining to the representation for the solution of the system of the higher-order rational difference equations xn+1 = 1/6−yn−k, yn+1 = 1/6−xn−k, n, k ∈ N0. where N0 = N ∪ {0}, and the initial conditions x−k, x−k+1, . . ., x0, y−k, y−k+1, . . ., y0 are non zero real numbers such that their solution is related to Balancing numbers. We also study the stability character and asymptotic behavior of this system. [ABSTRACT FROM AUTHOR]

In this paper, we construct and formulate the solutions and periodicity character of the following nonlinear rational systems of difference equations: S n + 1 = T n S n − 2 S n − 2 + T n − 1 , T n + 1 = S n T n − 2 ± T n − 2 ± S n − 1 , n = 0 , 1 , 2 ,.... , (0.1) where the initial conditions s − 2 , s − 1 , s 0 , t − 2 , t − 1 , t 0 are positive real numbers. Moreover, some mathematical programs are used to support our theoretical results of each system in this paper. [ABSTRACT FROM AUTHOR]

This paper aims to derive analytical expressions for solutions of fractional bidimensional systems of difference equations with higher-order terms under specific parametric conditions. Additionally, formulations of solutions for one-dimensional equations derived from these systems are explored. Furthermore, rigorous proof is provided for the local stability of the unique positive equilibrium point of the proposed systems. The theoretical findings are validated through numerical examples using MATLAB, facilitating graphical illustrations of the results.

In this paper, we represent that the following three-dimensional system of difference equations x n + 1 = α y n + a y n y n − β z n − 1 , y n + 1 = β z n + b z n z n − γ x n − 1 , z n + 1 = γ x n + c x n x n − α y n − 1 , n ∈ ℕ 0 , where the parameters a, b, c, α, β, γ and the initial values x−i, y−i, z−i, i ∈ {0, 1}, are real numbers, can be solved in closed form by using transformation. We analyzed the solutions in 10 different cases depending on whether the parameters are zero or nonzero. It is noteworthy to depict that the solutions of some particular cases of this system are presented in terms of generalized Fibonacci numbers. Note that our results considerably extend and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we derive the well-defined solutions to a θ-dimensional system of difference equations. We show that, the well-defined solutions to that system are represented in terms of Fibonacci and Lucas sequences. Moreover, we study the global stability of the solutions to that system. Finally, we give some numerical examples which confirm our theoretical results.

In this paper, we obtain the form of the solutions of the following rational systems of difference equations $ \begin{equation*} x_{n+1} = \dfrac{y_{n-1}z_{n}}{z_{n}\pm x_{n-2}}, \;y_{n+1} = \dfrac{z_{n-1}x_{n} }{x_{n}\pm y_{n-2}}, \ z_{n+1} = \dfrac{x_{n-1}y_{n}}{y_{n}\pm z_{n-2}}, \end{equation*} $ with initial values are non-zero real numbers.

We present thirty-six classes of three-dimensional systems of difference equations of the hyperbolic-cotangent type which are solvable in closed form.

Our aim in this paper is to obtain formulas for solutions of rational difference equations such as x n + 1 = 1 ± x n − 1 y n / 1 − y n , y n + 1 = 1 ± y n − 1 x n / 1 − x n , and x n + 1 = 1 ± x n − 1 y n − 2 / 1 − y n , y n + 1 = 1 ± y n − 1 x n − 2 / 1 − x n , where the initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , y 0 are non-zero real numbers. In addition, we show that the some of these systems are periodic with different periods. We also verify our theoretical outcomes at the end with some numerical applications and draw it by using some mathematical programs to illustrate the results. [ABSTRACT FROM AUTHOR]

In the present paper, we prove the boundedness of the positive solutions as well as local and global asymptotic stable of the equilibrium points of difference equations system in the second order Un+1=b1Vn+c1VnUn−1r+Vn+Un−1,Vn+1=b2Un+c2UnVn−1r+Un±Vn−1,$$ {\displaystyle \begin{array}{ll}{U}_{n+1}& ={b}_1{V}_n+\frac{c_1{V}_n{U}_{n-1}}{r+{V}_n+{U}_{n-1}},\\ {}{V}_{n+1}& ={b}_2{U}_n+\frac{c2{U}_n{V}_{n-1}}{r+{U}_n\pm {V}_{n-1}},\end{array}} $$ Furthermore, we study the solutions for three special cases of system. Numerical examples are provided to illustrate results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the dynamics of the system of two rational difference equations: xn+1 = αn + yn−k/ yn, yn+1 = αn + xn−k / xn, n = 0, 1, . . . where {αn}n≥0 is a two periodic sequence of nonnegative real numbers and the initial conditions xi, yi are arbitrary positive numbers for i = −k, −k + 1, −k + 2, . . ., 0 and k ∈ N. We investigate the boundedness character of positive solutions. In addition, we establish some sufficient conditions under which the local asymptotic stability and the global asymptotic stability are assured. Furthermore, we determine the rate of the convergence of the solutions. Some numerical are considered in order to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

We present the following new class of three‐dimensional systems of difference equations xn+3=1a2+a1yn+2+a0yn+2zn+1,yn+3=1b2+b1zn+2+b0zn+2xn+1,zn+3=1c2+c1xn+2+c0xn+2yn+1,n∈ℕ0,where aj,bj,cj,x1,x2,y1,y2,z1,z2∈ℝ,j=0,2‾, and show that the systems are solvable in closed form, considerably extending some results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the form of the solutions of the following systems of difference equations of second order xn+1xn+1==xnyn−1xn+yn,yn+1=xn−1ynxn+yn,xnyn−1xn−yn,yn+1=xn−1ynxn−yn, n=0,1,...,xn+1=xnyn−1xn+yn,yn+1=xn−1ynxn+yn,xn+1=xnyn−1xn−yn,yn+1=xn−1ynxn−yn, n=0,1,..., where the initial conditions x−1, x0, y−1 x−1, x0, y−1 and y0 y0 are arbitrary nonzero real numbers.