A Study on a Solvable Two-Dimensional System of Fourth-Order Difference Equations

Difference equations play a crucial role in modeling discrete dynamical systems across a wide range of scientific disciplines, including biology, economics, engineering, and physics. In this article, we examine a nonlinear system of rational difference equations of order four, defined by ? ?? ?? xn+1 = ynyn?3 axn?2 ?byn?3 , yn+1 = xnxn?3 cyn?2 ?dyn?3 , n ? N0, where a,b, c, and d are real parameters. The system is initialized with nonzero real values for x?3, x?2, x?1, x0, y?3, y?2, y?1, y0. The objectives of this study are to derive explicit closed-form solutions, determine conditions for the existence and uniqueness of solutions, and analyze the qualitative behavior of the system, including stability and asymptotic properties. The results offer new insights into the intricate dependence of the system’s dynamics on both the parameters and the initial conditions. This contributes to a deeper understanding of high-order nonlinear difference systems and their complex behaviors.