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.

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 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 Global dynamics of some system of second-order difference equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tran Hong Thai ◽  
Nguyen Anh Dai ◽  
Pham Tuan Anh
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Global Dynamics ◽  
Solution Existence ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Boundedness And Persistence ◽  
Invariant Rectangle ◽  
Theoretical Results

<p style='text-indent:20px;'>In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-x_{n-1}}}{\gamma_1+y_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-y_{n-1}}}{\gamma_2+x_n},\\ x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-y_{n-1}}}{\gamma_1+x_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-x_{n-1}}}{\gamma_2+y_n}, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where the parameters <inline-formula><tex-math id="M1">\begin{document}$ \alpha_i,\ \beta_i,\ \gamma_i $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ i \in \{1,2\} $\end{document}</tex-math></inline-formula> and the initial conditions <inline-formula><tex-math id="M3">\begin{document}$ x_{-1}, x_0, y_{-1}, y_0 $\end{document}</tex-math></inline-formula> are positive real numbers. Some numerical example are given to illustrate our theoretical results.</p>

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.

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.

ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS OF DIFFERENCE EQUATION SYSTEM OF EXPONENTIAL FORM

Fractals ◽  
2020 ◽  
Vol 28 (06) ◽  
pp. 2050118
Author(s):  
ABDUL KHALIQ ◽  
MUHAMMAD ZUBAIR ◽  
A. Q. KHAN
Keyword(s):  
Initial Conditions ◽  
Equation System ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Exponential Form ◽  
Numerical Examples ◽  
Rational Difference Equations ◽  
Boundedness Character ◽  
Theoretical Results

In this paper, we study the boundedness character and persistence, local and global behavior, and rate of convergence of positive solutions of following system of rational difference equations [Formula: see text] wherein the parameters [Formula: see text] for [Formula: see text] and the initial conditions [Formula: see text] are positive real numbers. Some numerical examples are given to verify our theoretical results.

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.

scholarly journals Dynamics of a Higher-Order System of Difference Equations

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.

scholarly journals Analytical Study of a System of Dierence Equation

2019 ◽  
pp. 1-18
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed
Keyword(s):  
Analytical Study ◽  
Initial Conditions ◽  
Qualitative Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Predator Prey ◽  
Rational Difference Equations ◽  
Predator Prey Model ◽  
Special Cases ◽  
Theoretical Results

We study the qualitative behavior of a predator-prey model, where the carrying capacity of the predators environment is proportional to the number of prey. The considered system is given by the following rational difference equations: where the initial conditions x-2; x-1; x0; y-2; y-1; y0 are arbitrary positive real numbers. Also, we give specic form of the solutions of some special cases of this equation. Some numerical examples are given to verify our theoretical results.

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