On explicit periodic solutions in three-dimensional difference systems

This paper focuses on the existence and analytical formulation of closed-form solutions for a three-dimensional system of nonlinear difference equations. The proposed system possesses a mathematical architecture that encapsulates complex nonlinear interactions among three mutually dependent variables. Through the application of systematic analytical transformations, the original system was reduced to a set of solvable recurrence relations, thereby allowing the derivation of explicit closed-form expressions with a high degree of analytical precision. Furthermore, numerical examples revealed that even minute perturbations in the system parameters or initial conditions can induce significant variations in oscillatory patterns, highlighting the system's structural sensitivity and rich dynamical diversity. This paper, therefore, constitutes a natural and essential extension of previously studied two-dimensional frameworks toward more sophisticated three-dimensional models, which provide a more faithful representation of interdependent relationships within discrete nonlinear systems.