Solvability of a system of higher order nonlinear difference equations

Abstract

In this paper we show that the system of difference equations x n = ay n−k + dy n−k x n−(k+l) bx n−(k+l) + cy n−l , y n = αx n−k + δx n−k y n−(k+l) βy n−(k+l) + γx n−l , where n ∈ N 0 , k and l are positive integers, the parameters a, b, c, d, α, β, γ, δ are real numbers and the initial values x −j , y −j , j = 1, k + l, are real numbers, can be solved in the closed form. We also determine the asymptotic behavior of solutions for the case l = 1 and describe the forbidden set of the initial values using the obtained formulas. Our obtained results significantly extend and develop some recent results in the literature.

Stabilizing instability‐suboptimality‐and‐chaos‐prone fluctuations at crisis junctures: Stochastic possibilities for crisis management

“…Clearly, the trajectory above shows that fluctuations in foreign exchange are within a band and hence the policy, if it can be sustainably maintained, could prevent the market from sliding into an uncontrollably erratic path that could be induced by stochastic and chaotic tendencies. (For different modelling exercises that could be used for the purpose of developing alternative stochastic policies, see Kara (2007Kara ( , 2013. )…”

On a Solvable System of Non-Linear Difference Equations With Variable Coefficients

“…Recently, in [27] we showed that the following difference equations system     ) can obtain by taking 2 k  in system (1.5). Finally, we showed that the following higher-order system of nonlinear difference equations…”

Solutions formulas for some general systems of difference equations

“…It is clear that if we want to understand our models, we need to know the behavior of the solutions of the equations of the models, and this fact will be possible if we can solve in closed form these equations. One can find in the literature a lot of works on difference equations where explicit formulas of the solutions are given, see for instance [1], [2], [5], [8], [9], [10], [7], [12], [11], [13], [16], [15], [14], [17], [18], [21], [20], [22]. Such type of difference equations and systems is called solvable difference equations.…”