Solvability of Three-Dimensional Nonlinear Difference Systems via Transformations and Generalized Fibonacci Recursions

This paper presents closed-form solutions for a three-dimensional system of nonlinear difference equations with variable coefficients. The approach employs functional transformations and leverages generalized Fibonacci sequences to construct the solutions explicitly. These solutions reveal a profound connection to generalized Fibonacci recursions. The proposed method is based on sophisticated mathematical transformations that reduce the complex nonlinear system to a solvable linear form, followed by the derivation of general solutions through iterative techniques and harmonic analysis. Furthermore, we extend our results to a generalized class of systems by introducing flexible functional transformations, while rigorously maintaining the required regularity conditions. The findings demonstrate the effectiveness of this methodology in addressing a broad class of complex nonlinear systems and open new perspectives for modeling multivariate dynamical phenomena. The analysis further reveals two distinct dynamical regimes—an unbounded oscillatory growth phase and a bounded cyclic equilibrium—arising from the relative magnitude of the variable coefficients, thereby highlighting the method’s capacity to characterize both amplifying and self-regulating behaviors within a unified analytical framework.