System Of Rational Difference Equations Research Articles

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.

Global Dynamics of Higher-Order Exponential Systems of Difference Equations

In this paper, we study the global dynamics of three higher-order exponential systems of rational difference equations. This suggested work considerably extends and improves some existing results in the literature.

Explicit Solutions and Bifurcations for a System of Rational Difference Equations

In this paper, we consider the explicit solution of the following system of nonlinear rational difference equations: x n + 1 = x n - 1 / x n - 1 + r , y n + 1 = x n - 1 y n / x n - 1 y n + r , with initial conditions x - 1 , x 0 and y 0 , which are arbitrary positive real numbers. By doing this, we encounter the hypergeometric function. We also investigate global dynamics of this system. The global dynamics of this system consists of two kind of bifurcations.

PERIODICITY AND SOLUTIONS OF SOME RATIONAL DIFFERENCE EQUATIONS 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. Moreover, we study the periodicity of solutions for such systems. Finally, some numerical examples are presented.

The Form of the Solutions of System of Rational Difference Equation

In this article, we study the form of the solutions of the system of difference equations $x_{n+1}=((y_{n-8})/(1+y_{n-2}x_{n-5}y_{n-8}))$, $y_{n+1}=((x_{n-8})/(\pm1\pm x_{n-2}y_{n-5}x_{n-8}))$, with the initial conditions are real numbers. Also, we give the numerical examples of some of difference equations and got some related graphs and figures using by Matlab.

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.

On some rational systems of difference equations

On some rational systems of difference equations

Bifurcation and Global Dynamics of a Leslie-Gower Type Competitive System of Rational Difference Equations with Quadratic Terms

We investigate global dynamics of the following systems of difference equations xn+1=xn/A1+B1xn+C1yn, yn+1=yn2/A2+B2xn+C2yn2, n=0,1,…, where the parameters A1, A2, B1, B2, C1, and C2 are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.

Asymptotic behaviour of solutions of rational difference systems

In this article, we investigate the asymptotic behaviour of solutions of systems of rational difference equations in arbitrary dimensions. We give conditions for the parameters ensuring that the positive solutions of the considered system are bounded, unbounded, increasing, decreasing, and convergent, respectively.

Dynamics and Behaviour of Some Rational Systems of Difference Equations

Dynamics and Behaviour of Some Rational Systems of Difference Equations

Solution for Rational Systems of Difference Equations of Order Three

In this paper, we consider the solution and periodicity of the following systems of difference equations: x n + 1 = y n − 2 − 1 + y n − 2 x n − 1 y n , y n + 1 = x n − 2 ± 1 ± x n − 2 y n − 1 x n , with initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , and y 0 are nonzero real numbers.

On a solvable for some systems of rational difference equations

On a solvable for some systems of rational difference equations

Global Character of a Six-Dimensional Nonlinear System of Difference Equations

The aim of this paper is to study the dynamical behavior of positive solutions for a system of rational difference equations of the following form:un+1=αun-1/β+γvn-2p,vn+1=α1vn-1/β1+γ1un-2p,n=0,1,…, where the parametersα,β,γ,α1,β1,γ1,pand the initial valuesu-i,v-ifori=0,1,2are positive real numbers.

Behavior and periodic solutions of a two-dimensional systems of rational difference equations

This paper is devoted to investigate the local asymptotic stability, boundedness and periodic solutions of particular cases of the following general system of difference equations: x_{n+1}=\frac{a_{1}y_{n-1}+a_{2}x_{n-3}+a_{3}}{a_{4}y_{n-1}x_{n-3}+a_{5}},\text{ }y_{n+1}=\frac{b_{1}x_{n-1}+b_{2}y_{n-3}+b_{3}}{b_{4}x_{n-1}y_{n-3}+b_{5}}, where the initial conditions $x_{-3},$ $x_{-2},$ $x_{-1},$ $x_{0},$ $y_{-3},$ $y_{-2},$ $y_{-1}$ and $y_{0}$ are arbitrary nonzero real numbers and $a_{i}$ and $b_{i}$ for $i=1,2,3,4,5$ are arbitrary real numbers. Also, we give some numerical examples to illustrate our results.

The form of solutions and periodicity for some systems of third‐order rational difference equations

In this paper, we deal with the system that has solutions and the periodicity character of the following systems of rational difference equations with order three urn:x-wiley:mma:media:mma3547:mma3547-math-0001 with initial conditions x−2,x−1,x0,y−2,y−1, and y0 that are arbitrary nonzero real numbers. Some numerical examples will be given to illustrate our results. Copyright © 2015 John Wiley & Sons, Ltd.

Dynamics of a three‐dimensional systems of rational difference equations

We investigate in this paper the solutions and the periodicity of the following rational systems of difference equations of three‐dimensional urn:x-wiley:mma:media:mma3540:mma3540-math-0001 with initial conditions x−2,x−1,x0,y−2,y−1,y0,z−2,z−1andz0 are nonzero real numbers. Copyright © 2015 John Wiley & Sons, Ltd.

Some General Systems of Rational Difference Equations

We investigate behavior of solutions of the following systems of rational difference equations: xn+1=yn-3k-1/(±1±yn-(3k-1)xn-(2k-1)yn-(k-1)),yn+1=xn-3k-1/(±1±xn-3k-1yn-2k-1xn-k-1), where k is a positive integer and the initial conditions are real numbers. We show that every solution is periodic with 6k period, considerably improving the results in the literature.

Global dynamics of some systems of rational difference equations

In this paper, we study the qualitative behavior of some systems of second-order rational difference equations. More precisely, we study the equilibrium points, local asymptotic stability of equilibrium point, unstability of equilibrium points, global character of equilibrium point, periodicity behavior of positive solutions and rate of convergence of positive solutions of these systems. Some numerical examples are given to verify our theoretical results.