On the Solutions of Systems of Difference Equations

We study the global behaviour of the solutions of cyclic systems of q order k difference equations with q unknown sequences n 7 ! u ðiÞ n ; 1 # i # q. If s denotes the direct cyclic permutation ð1; 2; . . . ; qÞ 7 ! ð2; 3; . . . ; q; 1Þ, such a system is:for values of k and choices of f which correspond, first to cases of classical simple difference equations u nþk ¼ f ðu nþk21 ; u nþk22 ; . . . ; u n Þ: order 2 or 3 extended Lyness' equations, order 2 generalized Lyness' type equations associated to conics, or to elliptic cubics or quartics; and then with f which becomes from other more general order k equations.

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“…DðiÞ is nothing but F q . So, the result of global periodicity of [14] is also very easy to find again with our geometrical point of view.…”

Section: The Geometrical Methods and Its First Applications To System (3)mentioning

confidence: 75%

Global behaviour of solutions of cyclic systems ofqorder 2 or 3 generalized Lyness' difference equations and of other more general equations of higher order

Bastien

Rogalski

2011

Journal of Difference Equations and Applications

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“…. , (25) with φ, ψ > 0. Furthermore, every solution of system (5) converges to a period-two solution.…”

Section: Periodicitymentioning

confidence: 99%

On a System of k-Difference Equations of Order Three

Yalçınkaya

Ahmad

Tollu

et al. 2020

Mathematical Problems in Engineering

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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.

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“…Kurbanli et al [9] investigated the positive solutions of the system of difference equations x n+1 = x n−1 /(y n x n−1 + 1), y n+1 = y n−1 /(x n y n−1 + 1). Yalcinkaya et al [10] investigated the sufficient conditions for the global asymptotic stability of the following system [11] studied many types of bifurcations for systems of several systems of rational difference equations. Many papers deal with the difference equations system.…”

Section: Introductionmentioning

confidence: 99%

Explicit Solutions and Bifurcations for a System of Rational Difference Equations

2019

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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.

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On the Solutions of Systems of Difference Equations

Cited by 37 publications

References 7 publications

Global behaviour of solutions of cyclic systems ofqorder 2 or 3 generalized Lyness' difference equations and of other more general equations of higher order

Global behaviour of solutions of cyclic systems ofqorder 2 or 3 generalized Lyness' difference equations and of other more general equations of higher order

On a System of k-Difference Equations of Order Three

Explicit Solutions and Bifurcations for a System of Rational Difference Equations

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