From Bilinear Recursions to Explicit Solutions: A Study of a Two-Dimensional Difference Equation System

This paper presents an analytical investigation of a class of two-dimensional nonlinear discrete systems governed by bilinear difference equations. By leveraging the explicit representation theory for first-order bilinear recursions, we derive closed-form representations for the solution sequences under general assumptions on system parameters. Special attention is given to parameter configurations that lead to structural simplifications, such as symmetry-induced reductions to equivalent one-dimensional dynamics, thereby yielding more transparent analytical expressions. Several numerical examples are presented to illustrate the rich dynamical behavior of the proposed system, including oscillatory patterns and the interplay between its components.