scholarly journals Dynamics of a second-order nonlinear difference system with exponents

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
D. S. Dilip ◽  
Smitha Mary Mathew
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Second Order ◽  
Difference System ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference

AbstractIn this paper, we study the persistence, boundedness, convergence, invariance and global asymptotic behavior of the positive solutions of the second-order difference system $$\begin{aligned} x_{n+1}&= \alpha _1 + a e ^{-x_{n-1}} + b y_{n} e ^{-y_{n-1}},\\ y_{n+1}&= \alpha _2 +c e ^{-y_{n-1}}+ d x_{n} e ^{-x_{n-1}} \quad n=0,1,2,\ldots \end{aligned}$$ x n + 1 = α 1 + a e - x n - 1 + b y n e - y n - 1 , y n + 1 = α 2 + c e - y n - 1 + d x n e - x n - 1 n = 0 , 1 , 2 , … where $$\alpha _1, \alpha _2, a, b , c,d$$ α 1 , α 2 , a , b , c , d are positive real numbers and the initial conditions $$x_{-1},x_0, y_{-1}, y_0$$ x - 1 , x 0 , y - 1 , y 0 are arbitrary nonnegative 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.

On some difference equations of exponential form

2018 ◽  
pp. 581-584
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Difference Equations ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Exponential Form ◽  
Numerical Examples

In this paper, the local asymptotic behavior of positive solutions of some exponential difference equations x_(n+1)=(x_n+x_(n-k))/(1+x_(n-k) e^(x_(n-k) ) ) , k ∈ N, n=0,1,2,… are investigated where the initial conditions are arbitrary positive real numbers. Furthermore, some numerical examples are presented to verify our 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.

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 Stability of Nonhyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
S. Atawna ◽  
R. Abu-Saris ◽  
E. S. Ismail ◽  
I. Hashim
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
Hyperbolic Equilibrium

This is a continuation part of our investigation in which the second order nonlinear rational difference equation xn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1), n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0, is considered. The first part handled the global asymptotic stability of the hyperbolic equilibrium solution of the equation. Our concentration in this part is on the global asymptotic stability of the nonhyperbolic equilibrium solution of the equation.

scholarly journals Stability of Hyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-21 ◽  
Author(s):  
S. Atawna ◽  
R. Abu-Saris ◽  
E. S. Ismail ◽  
I. Hashim
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
Hyperbolic Equilibrium

Our goal in this paper is to investigate the global asymptotic stability of the hyperbolic equilibrium solution of the second order rational difference equation xn+1=α+βxn+γxn-1/A+Bxn+Cxn-1, n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0. In particular, we solve Conjecture 5.201.1 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.2 in Kulenović and Ladas monograph (2002).

scholarly journals On the Recursive Sequencexn=A+xn−kp/xn−1r

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Fangkuan Sun ◽  
Xiaofan Yang ◽  
Chunming Zhang
Keyword(s):  
Difference Equation ◽  
Dynamic Behavior ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
The Difference ◽  
The Stability

This paper studies the dynamic behavior of the positive solutions to the difference equationxn=A+xn−kp/xn−1r,n=1,2,…, whereA,p, andrare positive real numbers, and the initial conditions are arbitrary positive numbers. We establish some results regarding the stability and oscillation character of this equation forp∈(0,1).

On the solutions of a second-order difference equation in terms of generalized Padovan sequences

2018 ◽  
Vol 68 (3) ◽  
pp. 625-638 ◽  
Author(s):  
Yacine Halim ◽  
Julius Fergy T. Rabago
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Initial Conditions ◽  
Solution Stability ◽  
Two Dimensional ◽  
Real Numbers ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractThis paper deals with the solution, stability character and asymptotic behavior of the rational difference equation$$\begin{array}{} \displaystyle x_{n+1}=\frac{\alpha x_{n-1}+\beta}{ \gamma x_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{array}$$where ℕ0= ℕ ∪ {0},α,β,γ∈ ℝ+, and the initial conditionsx–1andx0are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by$$\begin{array}{} \displaystyle x_{n+1} = \frac{\alpha x_{n-1} + \beta}{\gamma y_n x_{n-1}}, \qquad y_{n+1} = \frac{\alpha y_{n-1} +\beta}{\gamma x_n y_{n-1}} ,\qquad n\in \mathbb{N}_0. \end{array}$$

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 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.

Sign in / Sign up

Export Citation Format

Share Document