On Some Solvable Systems of Some Rational Difference Equations of Third Order

Abstract

Our aim in this paper is to obtain formulas for solutions of rational difference equations such as xn+1=1±xn−1yn/1−yn,yn+1=1±yn−1xn/1−xn, and xn+1=1±xn−1yn−2/1−yn,yn+1=1±yn−1xn−2/1−xn, where the initial conditions x−2, x−1, x0, y−2, y−1, y0 are non-zero real numbers. In addition, we show that the some of these systems are periodic with different periods. We also verify our theoretical outcomes at the end with some numerical applications and draw it by using some mathematical programs to illustrate the results.

Analytical Study of Nonlinear Systems of Higher-Order Difference Equations: Solutions, Stability, and Numerical Simulations

“…Many researchers have made significant contributions to unraveling the behavior of solved difference equations and systems, shedding light on their dynamics and stability properties. For instance, Elsayed and Alshabi [14] delved into the solution forms and stability properties of second-order systems, while Al-Basyouni and Elsayed [15] provided formulas for solutions to systems of rational difference equations of various orders, demonstrating periodicity in certain cases. Okumuş and Soykan [16] extended this exploration to two-dimensional systems associated with Tribonacci numbers.…”

The Dynamics of a General Model of the Nonlinear Difference Equation and Its Applications

“…Recently, Elsayed et al [27][28][29], Al-Basyouni and Elsayed [30], and Kara and Yazlik [31] established solutions to for certain categories of DIEs. In [27], Elsayed and Alofi studied the properties of solutions to a system of DIEs and provided solutions to this system.…”

“…and provided solutions to this DIE. The periodic properties and construction of the solution for some rational system of DIEs were presented in [29,30]. Moreover, for fractional difference equations and systems, there are many interesting results in [32,33].…”