scholarly journals Global Attractor of Solutions of a Rational System in the Plane

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Miron B. Bekker ◽  
Martin J. Bohner ◽  
Hristo D. Voulov
Keyword(s):  
Positive Solution ◽  
Global Attractor ◽  
Difference Equations ◽  
Autonomous System ◽  
Additional Assumption ◽  
Finite Limit ◽  
Two Dimensional ◽  
Rational System ◽  
Rational Difference Equations

We consider a two-dimensional autonomous system of rational difference equations with three positive parameters. It was conjectured that every positive solution of this system converges to a finite limit. Here we confirm this conjecture, subject to an additional assumption.

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 Qualitative study of a third order rational system of difference equations

2021 ◽  
Vol 25 (1) ◽  
pp. 81-97
Author(s):  
Mehmet Gümüş ◽  
Raafat Abo-Zeid
Keyword(s):  
Qualitative Study ◽  
Rate Of Convergence ◽  
Positive Solutions ◽  
Difference Equations ◽  
System Of Difference Equations ◽  
Third Order ◽  
Rational System ◽  
Numerical Examples ◽  
Rational Difference Equations ◽  
Zero Equilibrium

This paper is concerned with the dynamics of positive solutions for a system of rational difference equations of the following form un+1 = au2 n-1 b + gvn-2 , vn+1 = a1v 2 n-1 b1 + g1un-2 , n = 0, 1, . . . , where the parameters a, b, g, a1, b1, g1 and the initial values u-i, v-i ∈ (0, ∞), i = 0, 1, 2. Moreover, the rate of convergence of a solution that converges to the zero equilibrium of the system is discussed. Finally, some numerical examples are given to demonstrate the effectiveness of the results obtained.

scholarly journals On the System of High Order Rational Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Qianhong Zhang ◽  
Wenzhuan Zhang ◽  
Yuanfu Shao ◽  
Jingzhong Liu
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Difference Equations ◽  
High Order ◽  
Rational Difference Equations

This paper is concerned with the boundedness, persistence, and global asymptotic behavior of positive solution for a system of two rational difference equations xn+1=A+(xn/∑i=1kyn-i),    yn+1=B+(yn/∑i=1kxn-i),  n=0,1,…,k∈{1,2,…}, where A,B∈0,∞,  x-i∈0,∞, and   y-i∈0,∞,  i=0,1,2,…,k.

scholarly journals On a Class of Nonautonomous Max-Type Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Wanping Liu ◽  
Xiaofan Yang ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Positive Sequence ◽  
Finite Limit

This paper addresses the max-type difference equationxn=max{fn/xn−kα,B/xn−mβ},n∈ℕ0, wherek,m∈ℕ,B>0, and(fn)n∈ℕ0is a positive sequence with a finite limit. We prove that every positive solution to the equation converges tomax{(limn→∞fn)1/(α+1),B1/(β+1)}under some conditions. Explicit positive solutions to two particular cases are also presented.

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.

Sign in / Sign up

Export Citation Format

Share Document