Some General Systems of Rational Difference Equations

General Systems

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.

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

Asian-European Journal of Mathematics ◽

10.1142/s1793557122500875 ◽

2021 ◽

pp. 2250087

Author(s):

Ibtissam Talha ◽

Salim Badidja

Keyword(s):

Difference Equations ◽

Initial Conditions ◽

Fibonacci Numbers ◽

General System ◽

Real 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.

Solutions of Some Difference Equations Systems and Periodicity

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v16i0.8057 ◽

2019 ◽

Vol 16 ◽

pp. 8247-8261

Author(s):

Elsayed M. Elsayed ◽

Faris Alzahrani ◽

Ibrahim Abbas ◽

N. H. Alotaibi

Keyword(s):

Difference Equations ◽

Initial Conditions ◽

Real Numbers ◽

Article Analysis

In this article, analysis and investigation have been conducted on the periodic nature as well as the type of the solutions of the subsequent schemes of rational difference equations with a nonzero real numbers initial conditions.

Dynamics of a Higher-Order System of Difference Equations

10.1155/2017/4312640 ◽

2017 ◽

Vol 2017 ◽

pp. 1-9

Author(s):

Qi Wang ◽

Qinqin Zhang ◽

Qirui Li

Keyword(s):

Positive Integer ◽

Positive Solutions ◽

Difference Equations ◽

Initial Conditions ◽

Higher Order ◽

Order System ◽

Real Numbers ◽

Precise Description ◽

System Of Difference Equations ◽

Positive Real

Consider the following system of difference equations:xn+1(i)=xn-m+1(i)/Ai∏j=0m-1xn-j(i+j+1)+αi,xn+1(i+m)=xn+1(i),x1-l(i+l)=ai,l,Ai+m=Ai,αi+m=αi,i,l=1,2,…,m;n=0,1,2,…,wheremis a positive integer,Ai,αi,i=1,2,…,m, and the initial conditionsai,l,i,l=1,2,…,m,are positive real numbers. We obtain the expressions of the positive solutions of the system and then give a precise description of the convergence of the positive solutions. Finally, we give some numerical results.

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

Mathematica Slovaca ◽

10.1515/ms-2017-0210 ◽

2019 ◽

Vol 69 (1) ◽

pp. 147-158 ◽

Cited By ~ 4

Author(s):

R. Abo-Zeid

Keyword(s):

Explicit Formula ◽

Difference Equations ◽

Initial Conditions ◽

Global Behavior ◽

Real Numbers ◽

Positive Real ◽

Third Order ◽

The Difference

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 $$\begin{array}{} \displaystyle x_{n+1}=\frac{ax_{n}x_{n-1}}{bx_{n-1}+ cx_{n-2}},\quad n=0,1,\ldots \end{array} $$ where a,b,c are positive real numbers and the initial conditions x−2,x−1,x0 are real numbers.

Explicit Solutions and Bifurcations for a System of Rational Difference Equations

Mathematics ◽

10.3390/math7010069 ◽

2019 ◽

Vol 7 (1) ◽

pp. 69

Author(s):

Bashir Al-Hdaibat ◽

Saleem Al-Ashhab ◽

Ramadan Sabra

Keyword(s):

Hypergeometric Function ◽

Difference Equations ◽

Explicit Solution ◽

Initial Conditions ◽

Global Dynamics ◽

Explicit Solutions ◽

Real Numbers ◽

Positive Real ◽

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.

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

10.1155/2018/1743540 ◽

2018 ◽

Vol 2018 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

A. M. Alotaibi ◽

M. S. M. Noorani ◽

M. A. El-Moneam

Keyword(s):

Difference Equations ◽

Initial Conditions ◽

Theoretical Discussion ◽

Real Numbers ◽

Nonlinear Difference Equations ◽

Positive Real ◽

Third Order ◽

Numerical Examples ◽

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.

Displaying the Structure of the Solutions for Some Fifth-Order Systems of Recursive Equations

Mathematical Problems in Engineering ◽

10.1155/2021/6682009 ◽

2021 ◽

Vol 2021 ◽

pp. 1-14

Author(s):

H. S. Alayachi ◽

A. Q. Khan ◽

M. S. M. Noorani ◽

A. Khaliq

Keyword(s):

Nonlinear Systems ◽

Numerical Simulations ◽

Difference Equations ◽

Initial Conditions ◽

Real Numbers ◽

Recursive Equations ◽

Fifth Order ◽

Theoretical Results

This paper presents the solutions to the following nonlinear systems of rational difference equations: x n + 1 = x n − 3 y n − 4 / y n 1 + x n − 1 y n − 2 x n − 3 y n − 4 , y n + 1 = y n − 3 x n − 4 / x n ± 1 ± y n − 1 x n − 2 y n − 3 x n − 4 where initial conditions x − δ , y − δ δ = 4,3 , … , 0 are nonnegative real numbers. Finally some numerical simulations are presented to verify obtained theoretical results.

The Form of the Solutions and Periodicity of Some Systems of Difference Equations

10.1155/2012/406821 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 5

Author(s):

M. Mansour ◽

M. M. El-Dessoky ◽

E. M. Elsayed

Keyword(s):

Difference Equations ◽

Initial Conditions ◽

Real Numbers ◽

This paper is devoted to get the form of the solutions and the periodic nature of the following systems of rational difference equationsxn+1=xn-5/(-1+xn-5yn-2), yn+1=yn-5/(±1±yn-5xn-2), where the initial conditions are real numbers.

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

Mathematica Slovaca ◽

10.1515/ms-2017-0378 ◽

2020 ◽

Vol 70 (3) ◽

pp. 641-656

Author(s):

Amira Khelifa ◽

Yacine Halim ◽

Abderrahmane Bouchair ◽

Massaoud Berkal

Keyword(s):

General Solution ◽

Closed Form ◽

Difference Equations ◽

Fibonacci Numbers ◽

Higher Order ◽

Real Numbers ◽

Lucas Numbers ◽

Initial Values ◽

Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

On the Solutions of the System of Difference Equationsxn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn}

10.1155/2009/325296 ◽

2009 ◽

Vol 2009 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

Dağistan Simsek ◽

Bilal Demir ◽

Cengiz Cinar

Keyword(s):

Difference Equations ◽

Initial Conditions ◽

Real Numbers ◽

System Of Difference Equations ◽

Positive Real

We study the behavior of the solutions of the following system of difference equationsxn+1=max⁡{A/xn,yn/xn},yn+1=max⁡{A/yn,xn/yn}where the constantAand the initial conditions are positive real numbers.