Order Rational Difference Equations Research Articles

Periodically and Stability Properties of a Higher Order Rational Difference Equation

The aim of this research is to study the local, global, and boundedness of the difference equationTη+1 = r + p1Tη−l1 / Tη−m1 + p2Tη−l1 / Tη−m2 + ... + psTη−l1 / Tη−ms,where l1, m1, m2, ..., ms, s, are positive real numbers. It also studies periodic solutions of special case of this equation. Finally, numerical examples are given to confirm results.

On a Solvable Systems of Third Order Rational Difference Equations

On a Solvable Systems of Third Order Rational Difference Equations

Global attractivity of a rational difference equation with higher order and its applications

<p>We study in this paper the global attractivity for a higher order rational difference equation. As application, our results not only include and generalize many known ones, but also formulate some new results for several conjectures presented by Camouzis and Ladas, et al.</p>

Qualitative study of a second order difference equation

In this paper, we study a second order rational difference equation. We analyze the stability of the unique positive equilibrium of the equation and prove the existence of a Neimark-Sacker bifurcation, validating our theoretical analysis via a numerical exploration of the system.

Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$

The main goal of this paper is to study the bifurcation of a second order rational difference equation $$x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+Cx_{n-1}}, ~~n=0, 1, 2, \ldots$$ with positive parameters $\alpha, \beta, A, B, C$ and non-negative initial conditions $\{x_{-k}, x_{-k+1}, \ldots, x_{0}\}$. We study the dynamic behavior and the direction of the bifurcation of the period-two cycle. Numerical discussion with figures are given to support our results.

Asymptotic behaviour of the solutions of a class of $(k+1)$-order rational difference equations

In this paper we investigate the solutions of a class of $(k+1)$-order rational difference equations. We give conditions for the parameters ensuring that the considered equation has a unique positive equilibrium point that is locally asymptotically stable and every positive solution of the considered equation is bounded and increasing and converges to the unique positive equilibrium point.

Periodic solutions and stability of eighth order rational difference equations

Some real life problems are modeled using difference equations. Extracting the exact solutions of such equations is an active topic for some scientists. This paper investigates the equilibrium points, stability, boundedness, periodicity, and some exact solutions for eighth order rational difference equations. The exact solutions are obtained using the iterations method. We also present some 2D figures to show the validity of the obtained results. The used methods can be applied for other nonlinear difference equations.

The solution and dynamic behaviour of some difference equations of seventh order

Nonlinear difference equations are mostly utilized to describe some natural phenomena. The exact solutions of some models cannot be sometimes extracted. Therefore, investigating the long behaviour of the model may give the future pattern of the respective problem. This article manifests the long behaviour of a seventh order rational difference equation with positive real coefficients. Through this paper, the local stability, global stability, boundedness are obtained analytically and some figures are illustrated to test the accuracy of the solutions. The used technique can be extended to be utilized in solving some higher order difference equations.

Dynamics and Bifurcation of A second Order Rational Difference Equation with Quadratic Terms

Dynamics and Bifurcation of A second Order Rational Difference Equation with Quadratic Terms

Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$

In this paper, we study dynamics and bifurcation of the third order rational difference equation \begin{eqnarray*} x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+Cx_{n-2}}, ~~n=0, 1, 2, \ldots \end{eqnarray*} with positive parameters $\alpha, \beta, A, B, C$ and non-negative initial conditions $\{x_{-k}, x_{-k+1}, \ldots, x_{0}\}$. We study the dynamic behavior, the sufficient conditions for the existence of the Neimark-Sacker bifurcation, and the direction of the Neimark-Sacker bifurcation. Then, we give numerical examples with figures to support our results.

Dynamics of Kth Order Rational Difference Equation

Dynamics of Kth Order Rational Difference Equation

On a fourth order rational difference equation

In this paper, we determine and study the behavior of all admissible solutions of the difference equation $$x_{n+1}=\frac{x_{n}x_{n-2}}{ax_{n-2}+ bx_{n-3}},\quad n=0,1,\ldots,$$ where $a,b$ are positive real numbers and the initial conditions $ x_{-3},x_{-2},x_{-1},x_0$ are real numbers. We show when $a=b=1$ that, every admissible solution converges to $0$.

Neimark-Sacker Bifurcation of a Third Order Difference Equation

In this paper, we investigate the bifurcation of a third order rational difference equation. Firstly, we show that the equation undergoes a Neimark-Sacker bifurcation when the parameter reaches a critical value. Then, we consider the direction of the Neimark-Sacker bifurcation. Finally, we give some numerical simulations of our results.

Global behavior of two third order rational difference equations with quadratic terms

Abstract In this paper, we determine the forbidden sets, introduce an explicit formula for the solutions and discuss the global behaviors of solutions of the difference equations x n + 1 = a x n x n − 1 b x n − 1 + c x n − 2 , n = 0 , 1 , … $$\begin{array}{} \displaystyle x_{n+1}=\frac{ax_{n}x_{n-1}}{bx_{n-1}+ cx_{n-2}},\quad n=0,1,\ldots \end{array} $$ where a,b,c are positive real numbers and the initial conditions x −2,x −1,x 0 are real numbers.

Behavior of solutions of a second order rational difference equation

Behavior of solutions of a second order rational difference equation

On the global dynamics of a rational difference equation with periodic coefficients

The aim of this paper is to investigate the qualitative behavior of a higher-order nonautonomous rational difference equation with periodic coefficients. Particularly, our investigation gives some answers to two open problems proposed by Camouzis and Ladas in their monograph (Dynamics of third order rational difference equations with open problems and conjectures. CRC, Boca Raton, 2008).

Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation

We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12, n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.

Dynamics of some higher order rational difference equations

Dynamics of some higher order rational difference equations

Neimark‐Sacker bifurcation of a fourth order difference equation

In this article, we study stability and bifurcation of a fourth order rational difference equation. We give condition for local stability, and we show that the equation undergoes a Neimark‐Sacker bifurcation. Moreover, we consider the direction of the Neimark‐Sacker bifurcation. Finally, we numerically validate our analytical results.

Dynamics of higher order rational difference equation $x_{n+1}=(alpha+beta x_{n})/(A + Bx_{n}+ Cx_{n-k})$

The main goal of this paper is to investigate the periodic character, invariant intervals, oscillation and global stability and other new results of all positive solutions of the equation$$x_{n+1}=frac{alpha+beta x_{n}}{A + Bx_{n}+ Cx_{n-k}},~~ n=0,1,2,ldots,$$where the parameters $alpha$, $beta$, $A$, $B$ and $C$ are positive, and the initial conditions $x_{-k},x_{-k+1},ldots,x_{-1},x_{0}$ are positive real numbers and $kin{1,2,3,ldots}$. We give a detailed description of the semi-cycles of solutions and determine conditions under which the equilibrium points are globally asymptotically stable. In particular, our paper is a generalization of the rational difference equation that was investigated by Kulenovic et al. [The Dynamics of $x_{n+1}=frac{alpha +beta x_{n}}{A+Bx_{n}+ C x_{n-1}}$, Facts and Conjectures, Comput. Math. Appl. 45 (2003) 1087--1099].