On the Behavior of Solutions of the System of Rational Difference Equations: , , and

In the proposed work, global dynamics of a 3×6 system of rational difference equations has been studied in the interior of R+3. It is proved that system has at least one and at most seven boundary equilibria and a unique +ve equilibrium under certain parametric conditions. By utilizing method of Linearization, local dynamical properties about equilibria have been investigated. It is shown that every +ve solution of the system is bounded, and equilibrium P0 becomes a globally asymptotically stable if α1<α2,α4<α5, α7<α8. It is also shown that every +ve solution of the system converges to P0. Finally theoretical results are verified numerically.

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Qualitative study of a third order rational system of difference equations

Mathematica Moravica ◽

10.5937/matmor2101081g ◽

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.

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Global Character of a Six-Dimensional Nonlinear System of Difference Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2016/6842521 ◽

2016 ◽

Vol 2016 ◽

pp. 1-7 ◽

Cited By ~ 4

Author(s):

Mehmet Gümüş ◽

Yüksel Soykan

Keyword(s):

Nonlinear System ◽

Positive Solutions ◽

Difference Equations ◽

Dynamical Behavior ◽

Real Numbers ◽

System Of Difference Equations ◽

Positive Real ◽

Rational Difference Equations ◽

Initial Values ◽

Global Character

The aim of this paper is to study the dynamical behavior of positive solutions for a system of rational difference equations of the following form:un+1=αun-1/β+γvn-2p,vn+1=α1vn-1/β1+γ1un-2p,n=0,1,…, where the parametersα,β,γ,α1,β1,γ1,pand the initial valuesu-i,v-ifori=0,1,2are positive real numbers.

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On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

Mathematica Slovaca ◽

10.1515/ms-2017-0378 ◽

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.

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Dynamics of positive solutions of a system of difference equations

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2021.113489 ◽

2021 ◽

Vol 392 ◽

pp. 113489

Author(s):

S. Abualrub ◽

M. Aloqeili

Keyword(s):

Positive Solutions ◽

Difference Equations ◽

System Of Difference Equations

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On the Solutions of the System of Difference Equationsxn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn}

Discrete Dynamics in Nature and Society ◽

10.1155/2009/325296 ◽

2009 ◽

Vol 2009 ◽

pp. 1-11 ◽

Cited By ~ 3

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.

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