Theoretical Solvability of a Three-Dimensional Cyclic Delayed Rational Difference System

This paper introduces a new three-dimensional cyclic rational difference system with mixed delays and establishes its theoretical solvability under suitable nondegeneracy and well-definedness assumptions. The main novelty lies in combining cyclic interactions with dual-delay rational dynamics within a unified framework that extends a classical scalar delayed rational difference equation. By constructing an appropriate change of variables, the proposed nonlinear system is transformed into an equivalent bilinear delayed system, from which explicit solution formulas are derived for well-defined solutions satisfying the imposed assumptions. The obtained results demonstrate that, under these structural conditions, the cyclic coupling and mixed delays do not preclude theoretical solvability. Several degenerate cases are also investigated, leading to reduced linear, affine, or Riccati-type delayed systems that remain theoretically solvable. Finally, it is shown that the classical one-dimensional delayed rational equation is recovered as a symmetric invariant subcase of the proposed model, thereby confirming that the latter extends the scalar equation within a unified three-dimensional framework.