System Of Rational Difference Equations Research Articles

On the solutions and behavior of rational systems of difference equations

In this paper, we obtain the expressions of the solutions of the following nonlinear systems of difference equations x_{n+1}=((x_{n-9})/(1+x_{n-9}y_{n-4})), y_{n+1}=((y_{n-9})/(±1±x_{n-4}y_{n-9})),n=0,1,..., where the initial conditions x₋₉, x₋₈, x₋₇, x₋₆, x₋₅, x₋₄, x₋₃, x₋₂, x₋₁, x₀, y₋₉, y₋₈, y₋₇, y₋₆, y₋₅, y₋₄, y₋₄, y₋₃, y₋₂, y₋₁, y₀ are arbitrary non zero real numbers.

Global stability and co-balancing numbers in a system of rational difference equations

<abstract><p>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.</p></abstract>

An investigation of the solutions for rational systems of difference equations of order two

An investigation of the solutions for rational systems of difference equations of order two

The periodic nature and expression on solutions of some rational systems of difference equations

Our aim in this paper is to obtain formulas expressions for solutions of the following fractional systems of difference equationsLn+1=Ln-3Mn-4Mn±1-Ln-3Mn-4Wn-1Xn-2,n=0,1,2,…,Mn+1=Mn-3Wn-4Wn±1-Mn-3Wn-4Xn-1Ln-2,Wn+1=Wn-3Xn-4Xn±1±Wn-3Xn-4Ln-1Mn-2,Xn+1=Xn-3Ln-4Ln±1±Xn-3Ln-4Mn-1Wn-2,where the initial values are non-zero real numbers. In addition we prove that some of these systems are periodic with different period. We also verify our theoretical results for every system with some numerical examples and draw by using some famous mathematical programs (Matlab program) to illustrate the results.

On a class of nonlinear rational systems of difference equations

<abstract><p>In this paper, we construct and formulate the solutions and periodicity character of the following nonlinear rational systems of difference equations:</p> <p><disp-formula> <label/> <tex-math id="FE38"> \begin{document} $ \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} $ \end{document} </tex-math> </disp-formula></p> <p>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.</p></abstract>

The expressions and behavior of solutions for nonlinear systems of rational difference equations

In this paper, we investigate the form of the solutions of two systems of rational difference equations of second order, where the initial conditions are arbitrary nonzero real numbers.

The form of the solutions of fourth order rational systems of difference equations

In this paper, we get the form of the solutions of the following difference equation systems of order four where the initial conditions z−3, z−2, z−1, z0, w−3, w−2, w−1, w0 are arbitrary nonzero real numbers.

Bifurcation and Chaos Control of a System of Rational Difference Equations

We study a system of rational difference equations in this article. For equilibrium points, we present the stability conditions. In addition, we show that the system encounters period-doubling bifurcation at the trivial equilibrium point O and Neimark-Sacker bifurcation at the non-trivial equilibrium point E. To control the chaotic behavior of the system, we use the hybrid control approach. We also verify our theoretical outcomes at the end with some numerical applications

On periodicity of systems of rational difference equations

In this paper, we investigate the periodicity of two systems of rational sequences of second and third order, respectively. The systems include a permutation that gives the ability of changing the appearance of components of solutions in the equations of the systems. We find periods of systems in terms of the order of the permutation. The periodicity of two more systems of maximum type are studied. Finally, many illustrative examples are given.

Periodicity of the solutions of general system of rational difference equations related to fibonacci numbers

In this paper, we deal with the periodicity of solutions of the following general system rational of difference equations: [Formula: see text] where [Formula: see text] [Formula: see text] and the initial conditions are arbitrary nonzero real numbers.

May’s Host–Parasitoid geometric series model with a variable coefficient

During the last fifteen years there has been a significant growth of mathematical and scientific interest related to the study of difference equations with variable coefficients. The reason might as well be that variable coefficients make several scientific models more realistic. The question, usually asked, is about the effect on the dynamics of a difference equation, after imposing variability on one or more of the coefficients of the equation. In this article, we investigate the boundedness and in some cases, the global behavior of solutions of a system of rational difference equations with a variable coefficient. If the coefficient is constant, then, up to an appropriate change of variables, the system that we study isthe geometric series case of May’s Host–Parasitoid Model.

Form of the periodic solutions of some systems of higher order difference equations

In this paper, we investigate the periodic nature of the solutions of some systems of rational difference equations where the initial values are arbitrary real numbers such that the denominator is always nonzero. Furthermore, some numerical examples are presented to verify our theoretical results.

On the Solutions of Three-Dimensional Rational Difference Equation Systems

In this paper, we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case as a special case of the following system: x n + 1 = y n z n − 1 / y n ± x n − 2 , y n + 1 = z n x n − 1 / z n ± y n − 2 , and z n + 1 = x n y n − 1 / x n ± z n − 2 with initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , y 0 , z − 2 , z − 1 , and z 0 are nonzero real numbers. Moreover, we study some behavior of the systems such as the boundedness of solutions for such systems. Finally, we present some numerical examples by giving some numerical values for the initial values of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program MATLAB to confirm the obtained results.

On a solvable system of rational difference equations of higher order

In this paper, we present that the following system of difference equations $$ x_{n}=\frac{x_{n-k}z_{n-l}}{b_{n}x_{n-k}+a_{n}z_{n-k-l}}, \ y_{n}=\frac{y_{n-k}x_{n-l}}{d_{n}y_{n-k}+c_{n}x_{n-k-l}}, \ z_{n}=\frac{z_{n-k}y_{n-l}}{f_{n}z_{n-k}+e_{n}y_{n-k-l}}, $$ where $n\in \mathbb{N}_{0}$, $k,l\in\mathbb{N}$, the initial values $x_{-i},y_{-i},z_{-i}$ are real numbers, for $i \in \overline{1,k+l}$, and sequences $\left( a_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( b_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( c_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( d_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( e_{n}\right) _{n\in \mathbb{N}_{0}}$ and $\left( f_{n}\right) _{n\in \mathbb{N}_{0}}$ are non-zero real numbers, for all $n\in \mathbb{N}_{0}$, which can be solved in closed form. We describe the forbidden set of the initial values using the obtained formulas and also determine the asymptotic behavior of solutions for the case $k=3$, $l=1$, and the sequences $\left( a_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( b_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( c_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( d_{n}\right) _{n\in \mathbb{N}_{0}}$, $\left( e_{n}\right) _{n\in \mathbb{N}_{0}}$ and $\left( f_{n}\right) _{n\in \mathbb{N}_{0}}$ are constant. Our results considerably extend and improve some recent results in the literature.

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 = au2 n-1 b + gvn-2 , vn+1 = a1v 2 n-1 b1 + g1un-2 , n = 0, 1, . . . , where the parameters a, b, g, a1, b1, g1 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.

Dynamical properties of some rational systems of difference equations

We explore the equilibrium points, local and global dynamics, rate of convergence, instability and boundedness of positive solution of some rational systems of difference equations. As an application of difference equations in mathematical biology, we also explore the local dynamics about equilibrium points of the discrete‐time Levin's model. Finally, obtained results are verified numerically.

ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS OF DIFFERENCE EQUATION SYSTEM OF EXPONENTIAL FORM

In this paper, we study the boundedness character and persistence, local and global behavior, and rate of convergence of positive solutions of following system of rational difference equations [Formula: see text] wherein the parameters [Formula: see text] for [Formula: see text] and the initial conditions [Formula: see text] are positive real numbers. Some numerical examples are given to verify our theoretical results.

ON THE SYSTEMS OF RATIONAL DIFFERENCE EQUATIONS

ON THE SYSTEMS OF RATIONAL DIFFERENCE EQUATIONS

On a Competitive System of Rational Difference Equations

This paper aims to investigate the global stability and the rate of convergence of positive solutions that converge to the equilibrium point of the system of difference equations in the modeling competitive populations in the form $$ x_{n+1}^{(1)}=\frac{\alpha x_{n-2}^{(1)}}{\beta +\gamma \prod\limits_{i=0}^{2}x_{n-i}^{(2)}},\text{ }x_{n+1}^{(2)}=\frac{\alpha _{1}x_{n-2}^{(2)}}{\beta _{1}+\gamma _{1}\prod\limits_{i=0}^{2}x_{n-i}^{(1)} }\text{, }n=0,1,... $$ where the parameters $\alpha ,\beta ,\gamma ,\alpha _{1},\beta _{1},\gamma _{1}$ are positive numbers and the initial conditions $ x_{-i}^{(1)},x_{-i}^{(2)}$ are arbitrary non-negative numbers for $i\in \{0,1,2\}$.

Global Dynamics of a 3 × 6 System of Difference Equations

In the proposed work, global dynamics of a 3×6 system of rational difference equations has been studied in the interior of R+3. It is proved that system has at least one and at most seven boundary equilibria and a unique +ve equilibrium under certain parametric conditions. By utilizing method of Linearization, local dynamical properties about equilibria have been investigated. It is shown that every +ve solution of the system is bounded, and equilibrium P0 becomes a globally asymptotically stable if α1<α2,α4<α5, α7<α8. It is also shown that every +ve solution of the system converges to P0. Finally theoretical results are verified numerically.