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

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
M. Mansour ◽  
M. M. El-Dessoky ◽  
E. M. Elsayed
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Real Numbers ◽  
Rational Difference Equations

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.

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

2021 ◽  
pp. 2250087
Author(s):  
Ibtissam Talha ◽  
Salim Badidja
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
General System ◽  
Real Numbers ◽  
Rational Difference Equations

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.

scholarly journals Solutions of Some Difference Equations Systems and Periodicity

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 ◽  
Rational Difference Equations ◽  
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.

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

2019 ◽  
Vol 69 (1) ◽  
pp. 147-158 ◽  
Author(s):  
R. Abo-Zeid
Keyword(s):  
Explicit Formula ◽  
Difference Equations ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Third Order ◽  
Rational Difference Equations ◽  
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.

scholarly journals Explicit Solutions and Bifurcations for a System of Rational Difference Equations

Mathematics ◽  
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 ◽  
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.

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

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
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 ◽  
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.

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

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 ◽  
Rational Difference Equations ◽  
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.

scholarly journals Some General Systems of Rational Difference Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Ali Gelisken ◽  
Merve Kara
Keyword(s):  
Positive Integer ◽  
Difference Equations ◽  
Initial Conditions ◽  
Real Numbers ◽  
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.

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

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 ◽  
Rational Difference Equations ◽  
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.

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

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
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.

scholarly journals Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

2019 ◽  
pp. 1-12
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed ◽  
Faris Alzahrani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Analytical Study ◽  
Initial Conditions ◽  
Specific Form ◽  
Positive Real ◽  
Rational Difference Equations ◽  
Special Cases ◽  
The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

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