scholarly journals On the Dynamics of a Higher-Order Rational Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
A. M. Ahmed
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Higher Order ◽  
Real Numbers ◽  
Rational Difference Equation

The aim of this paper is to investigate the global asymptotic stability and the periodic character for the rational difference equationxn+1=αxn-1/(β+γΠi=lkxn-2ipi),  n=0,1,2,…, where the parametersα,β,γ,pl,pl+1,…,pkare nonnegative real numbers, andl,kare nonnegative integers such thatl≤k.

scholarly journals Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-22
Author(s):  
M. R. S. Kulenović ◽  
S. Moranjkić ◽  
M. Nurkanović ◽  
Z. Nurkanović
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Local Asymptotic Stability ◽  
Stable Periodic ◽  
Rational Difference Equation ◽  
Unknown Period

We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12,  n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.

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 Global Stability of a Rational Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Guo-Mei Tang ◽  
Lin-Xia Hu ◽  
Gang Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Stability ◽  
Positive Solutions ◽  
Open Problem ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Nonlinear Difference Equation ◽  
Real Numbers ◽  
Rational Difference Equation

We consider the higher-order nonlinear difference equation with the parameters, and the initial conditions are nonnegative real numbers. We investigate the periodic character, invariant intervals, and the global asymptotic stability of all positive solutions of the above-mentioned equation. In particular, our results solve the open problem introduced by Kulenović and Ladas in their monograph (see Kulenović and Ladas, 2002).

scholarly journals Dynamical Behavior of a System of Third-Order Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Qianhong Zhang ◽  
Jingzhong Liu ◽  
Zhenguo Luo
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Positive Solution ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Dynamical Behavior ◽  
Third Order ◽  
Rational Difference Equations ◽  
Rational Difference Equation

This paper deals with the boundedness, persistence, and global asymptotic stability of positive solution for a system of third-order rational difference equationsxn+1=A+xn/yn-1yn-2,yn+1=A+yn/xn-1xn-2,n=0,1,…, whereA∈(0,∞),x-i∈(0,∞);y-i∈(0,∞),i=0,1,2. Some examples are given to demonstrate the effectiveness of the results obtained.

scholarly journals Dynamics and Solutions’ Expressions of a Higher-Order Nonlinear Fractional Recursive Sequence

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Abeer Alshareef ◽  
Faris Alzahrani ◽  
Abdul Qadeer Khan
Keyword(s):  
Fixed Point ◽  
Difference Equation ◽  
Higher Order ◽  
Real Numbers ◽  
Stability Properties ◽  
Special Cases ◽  
Recursive Sequence ◽  
Rational Difference Equation

The principle purpose of this article is to examine some stability properties for the fixed point of the below rational difference equation U n + 1 = ξ U n − 8 + ε U n − 8 2 / μ U n − 8 + κ U n − 17 where ξ , ε , μ , and κ are arbitrary real numbers. Moreover, solutions for some special cases of the proposed difference equation are introduced.

scholarly journals Global Behavior of a New Rational Nonlinear Higher-Order Difference Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-4 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Nonnegative Integer ◽  
Equilibrium Solution ◽  
Higher Order ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Order Difference ◽  
Order Difference Equation

Let k be a nonnegative integer and c a real number greater than or equal to 1. We present qualitative global behavior of solutions to a rational nonlinear higher-order difference equation zn+1=(czn+zn-k+c-1znzn-k)/(znzn-k+c),  n≥0, with positive initial values z-k,z-k+1,⋯,z0, and show the global asymptotic stability of its positive equilibrium solution.

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