Uncovering Exact Solutions in Rational Difference Systems via Algebraic Transformations

This paper introduces a novel mathematical framework for studying a three-dimensional system of nonlinear difference equations, aimed at describing the discrete-time interactions among three interdependent variables. The proposed approach provides a systematic analytical treatment of the model and reveals its underlying dynamical structure. An innovative algebraic transformation is proposed to reformulate the original system into a factorable bilinear structure, which allows the derivation of explicit closed-form solutions in terms of the initial values and system parameters, without the need for iterative numerical procedures. The obtained solutions are classified according to the nature of the roots of the associated characteristic equation, leading to two distinct cases: repeated roots and distinct roots. Each case exhibits qualitatively different dynamical behaviors, reflecting the influence of the spectral properties of the system. Finally, numerical simulations are presented to illustrate and support the theoretical results, confirming the consistency between analytical derivations and computational behavior.