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 Behavior of a Competitive System of Second-Order Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Q. Din ◽  
T. F. Ibrahim ◽  
K. A. Khan
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Positive Equilibrium ◽  
Positive Real ◽  
Competitive System ◽  
Order Difference ◽  
Rational Difference Equations ◽  
Positive Equilibrium Point ◽  
Boundedness And Persistence ◽  
Theoretical Results

We study the boundedness and persistence, existence, and uniqueness of positive equilibrium, local and global behavior of positive equilibrium point, and rate of convergence of positive solutions of the following system of rational difference equations:xn+1=(α1+β1xn-1)/(a1+b1yn),yn+1=(α2+β2yn-1)/(a2+b2xn), where the parametersαi,βi,ai, andbifori∈{1,2}and initial conditionsx0,x-1,y0, andy-1are positive real numbers. Some numerical examples are given to verify our theoretical results.

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 Stability Analysis of a System of Exponential Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Q. Din ◽  
K. A. Khan ◽  
A. Nosheen
Keyword(s):  
Difference Equations ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Numerical Examples ◽  
Boundedness Character ◽  
Theoretical Results

We study the boundedness character and persistence, existence and uniqueness of positive equilibrium, local and global behavior, and rate of convergence of positive solutions of the following system of exponential difference equations:xn+1=(α1+β1e-xn+γ1e-xn-1)/(a1+b1yn+c1yn-1),yn+1=(α2+β2e-yn+γ2e-yn-1)/(a2+b2xn+c2xn-1), where the parametersαi, βi, γi, ai, bi, andcifori∈{1,2}and initial conditionsx0, x-1, y0, andy-1are positive real numbers. Furthermore, by constructing a discrete Lyapunov function, we obtain the global asymptotic stability of the positive equilibrium. Some numerical examples are given to verify our theoretical results.

scholarly journals On a System of k-Difference Equations of Order Three

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ibrahim Yalçınkaya ◽  
Hijaz Ahmad ◽  
Durhasan Turgut Tollu ◽  
Yong-Min Li
Keyword(s):  
Positive Solutions ◽  
Difference Equations ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Numerical Examples ◽  
Theoretical Results

In this paper, we deal with the global behavior of the positive solutions of the system of k -difference equations u n + 1 1 = α 1 u n − 1 1 / β 1 + α 1 u n − 2 2 r 1 ,   u n + 1 2 = α 2 u n − 1 2 / β 2 + α 2 u n − 2 3 r 2 , … ,   u n + 1 k = α k u n − 1 k / β k + α k u n − 2 1 r k , n ∈ ℕ 0 , where the initial conditions u − l i l = 0,1,2 are nonnegative real numbers and the parameters α i , β i , γ i , and r i are positive real numbers for i = 1,2 , … , k , by extending some results in the literature. By the end of the paper, we give three numerical examples to support our theoretical results related to the system with some restrictions on the parameters.

scholarly journals Analytic Solutions and Stability of Sixth Order Difference Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
H. S. Alayachi ◽  
M. S. M. Noorani ◽  
A. Q. Khan ◽  
M. B. Almatrafi
Keyword(s):  
Difference Equations ◽  
Local Stability ◽  
Initial Conditions ◽  
Fibonacci Sequence ◽  
Analytic Solutions ◽  
Sixth Order ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Order Difference Equation

In the present paper, the global attractor, local stability, and boundedness of the solution of sixth order difference equations are investigated analytically and numerically. The exact solutions of three equations are presented by utilizing Fibonacci sequence. We also analyse the periodicity of a sixth order difference equation. The considered difference equations are given by yn+1=Ayn−1±Byn−1yn−3/Cyn−3±Dyn−5, n=0,1,…, where the initial conditions y−5,y−4,y−3,y−2,y−1, and y0 are arbitrary real numbers and the values A,B,C, and D are defined as positive real 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 Boundedness of solutions and stability of certain second-order difference equation with quadratic term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Emin Bešo ◽  
Senada Kalabušić ◽  
Naida Mujić ◽  
Esmir Pilav
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Second Order ◽  
Quadratic Term ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractWe consider the second-order rational difference equation $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$xn+1=γ+δxnxn−12, where γ, δ are positive real numbers and the initial conditions $x_{-1}$x−1 and $x_{0}$x0 are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

scholarly journals Global Dynamics of Some Exponential Type Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-24
Author(s):  
A. Q. Khan ◽  
H. M. Arshad ◽  
B. A. Younis ◽  
KH. I. Osman ◽  
Tarek F. Ibrahim ◽  
...  
Keyword(s):  
Fixed Point ◽  
Lyapunov Function ◽  
Exponential Type ◽  
Type Systems ◽  
Global Dynamics ◽  
Dynamical Properties ◽  
Boundedness And Persistence ◽  
Invariant Rectangle ◽  
Theoretical Results ◽  
Positive Fixed Point

We explore the boundedness and persistence, existence of an invariant rectangle, local dynamical properties about the unique positive fixed point, global dynamics by the discrete-time Lyapunov function, and the rate of convergence of some 2,3-type exponential systems of difference equations. Finally, theoretical results are numerically verified.

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.

On the solutions of a higher order difference equation

2020 ◽  
Vol 27 (2) ◽  
pp. 165-175 ◽  
Author(s):  
Raafat Abo-Zeid
Keyword(s):  
Difference Equation ◽  
Explicit Formula ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Forbidden Set ◽  
Order Difference Equation ◽  
The Difference

AbstractIn this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equationx_{n+1}=\frac{ax_{n}x_{n-k}}{bx_{n}-cx_{n-k-1}},\quad n=0,1,\ldots,where{a,b,c}are positive real numbers and the initial conditions{x_{-k-1},x_{-k},\ldots,x_{-1},x_{0}}are real numbers. We show that when{a=b=c}, the behavior of the solutions depends on whetherkis even or odd.

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