Rational Difference Equations Research Articles

Qualitative Behavior of Two Rational Difference Equations

Obtaining the exact solutions of most rational recursive equations is sophisticated sometimes. Therefore, a considerable number of nonlinear difference equations is often investigated by studying the qualitative behavior of the governing forms of these equations. The prime purpose of this work is to analyse the equilibria, local stability, global stability character, boundedness character and the solution behavior of the following fourth order fractional difference equations: $$ x_{n+1}=\frac{\alpha x_{n}x_{n-3}}{\beta x_{n-3}-\gamma x_{n-2}},\ \ \ \ x_{n+1}=\frac{\alpha x_{n}x_{n-3}}{-\beta x_{n-3}+\gamma x_{n-2}},\ \ \ n=0,1,..., $$ where the constants $\alpha ,\ \beta ,\ \gamma \in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{+}$ and the initial values $x_{-3},\ x_{-2},\ x_{-1}$\ and $x_{0}$ are required to be arbitrary non zero real numbers. Furthermore, some numerical figures will be obviously shown in this paper.

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.

Note on the bilinear difference equation with a delay

A representation formula for the general solution to a higher‐order rational difference equation has been given recently in this journal. The formula was proved by the method of induction without giving any theoretical explanation related to it. Here we show how the corresponding representation formula for the general solution to a general higher‐order rational difference equation, that is, to the bilinear difference equation with a delay, is obtained in an elegant way. We also give some theoretical explanations related to the representation, as well as some explanations related to such types of difference equations. The corresponding representation for a system of bilinear difference equations with delay is also presented.

On the boundedness of solutions of a class of third-order rational difference equations

ABSTRACTWe consider the difference equation with and If we show that for all positive initial conditions the solutions of the difference equations are bounded. If then there exists positive initial conditions such that the solutions are unbounded.

Lie symmetry analysis and explicit formulas for solutions of some third-order difference equations

Lie group theory is applied to rational difference equations of the formwhere (an)n∈ℕ0, (bn)n∈ℕ0 are non-zero real sequences. Consequently, new symmetries are derived and exact solutions, in unified manner, are constructed. Based on some constraints in the expression of the symmetry generators, we split these solutions into different categories. This work generalises a recent result by Ibrahim [9].

Dynamics and Solutions of a Fifth-Order Nonlinear Difference Equation

The main objective of this paper is to study the behavior of the rational difference equation of the fifth-order yn+1=αyn+βynyn-3/(Ayn-4+Byn-3), n=0,1,…, where α,β,A, and B are real numbers and the initial conditions y-4,y-3,y-2,y-1 and y0 are positive real numbers such that Ay-4+By-3≠0. Also, we obtain the solution of some special cases of this equation.

Dynamics of some higher order rational difference equations

Dynamics of some higher order rational difference equations

Classification of global behavior of a system of rational difference equations

This paper deals with a system of rational difference equations xn+1=ayn+bcyn+d,yn+1=axn+bcxn+d,n=0,1,2,…,where a, b, c, d are real numbers with c≠0 and ad−bc≠0. We establish a representation formula of solutions of the system and classify global behavior of solutions when no initial values belong to the forbidden set of the system.

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.

Bifurcation Analysis and Chaos Control in a Second-Order Rational Difference Equation

AbstractThis work is related to dynamics of a second-order rational difference equation. We investigate the parametric conditions for local asymptotic stability of equilibria. Center manifold theorem and bifurcation theory are implemented to discuss the parametric conditions for existence and direction of period-doubling bifurcation and pitchfork bifurcation at trivial equilibrium point. Moreover, the parametric conditions for existence and direction of Neimark–Sacker bifurcation at positive steady state are investigated with the help of bifurcation theory. The chaos control in the system is discussed through implementation of OGY feedback control method. In particular, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control. Finally, numerical simulations are provided to illustrate theoretical results. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.

Global attractivity of a rational difference equation of order twenty

Global attractivity of a rational difference equation of order twenty

On the asymptotic character of a generalized rational difference equation

We investigate the global asymptotic stability of the solutions of \begin{document}$X_{n+1}=\frac{β X_{n-l} + γ X_{n-k}}{A + X_{n-k}} $\end{document} for \begin{document}$n=1,2, ...$\end{document} , where \begin{document}$l$\end{document} and \begin{document}$k$\end{document} are positive integers such that \begin{document}$l≠ k$\end{document} . The parameters are positive real numbers and the initial conditions are arbitrary nonnegative real numbers. We find necessary and sufficient conditions for the global asymptotic stability of the zero equilibrium. We also investigate the positive equilibrium and find the regions of parameters where the positive equilibrium is a global attractor of all positive solutions. Of particular interest for this generalized equation would be the existence of unbounded solutions and the existence of prime period two solutions depending on the combination of delay terms ( \begin{document}$l$\end{document} , \begin{document}$k$\end{document} ) being (odd, odd), (odd, even), (even, odd) or (even, even). In this manuscript we will investigate these aspects of the solutions for all such combinations of delay terms.

On the Solutions of a System of Third-Order Rational Difference Equations

The structure of the solutions for the system nonlinear difference equations xn+1=ynyn-2/(xn-1+yn-2), yn+1=xnxn-2/(±yn-1±xn-2), n=0,1,…, is clarified in which the initial conditions x-2, x-1, x0, y-2, y-1, y0 are considered as arbitrary positive real numbers. To exemplify the theoretical discussion, some numerical examples are presented.

On some rational difference equations of order eight

On some rational difference equations of order eight

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].

Forbidden sets and stability in some rational difference equations

In this paper, we solve open problem (5) submitted by Sedaghat in his paper, On third order rational difference equations with quadratic terms, J. Differ. Equ. Appl., 14(8) (2008), pp. 889–897. We also confirm conjecture (6) in the mentioned paper.

Qualitative study of a higher order rational difference equation

In this paper we study the behavior of the difference equation $x_{n+1}$ = $\dfrac{\alpha x_nx_{n-l}}{\beta x_{n-m}+\gamma x_{n-l}}$,$\quad n=0,1,$ $\cdots$ where the initial conditions $x_{-r}$, $x_{-r+1}$, $\cdots$ ,$x_0$ are arbitrary non zero real numbers where $r=\max\{l,m\}$ is a non-negative integer and $\alpha$, $\beta$ and $\gamma$ are constants: Also, we obtain the solutions of some special cases of this equation. At the end we present some numerical examples to support our theoretical discussion.

On the periodicity of a max-type rational difference equation

On the periodicity of a max-type rational difference equation

ON THE DYNAMICS OF A RATIONAL DIFFERENCE EQUATION

ON THE DYNAMICS OF A RATIONAL DIFFERENCE EQUATION