Structure and Explicit of Solutions for a Third-Order System of Rational Difference Equations

This paper investigates the qualitative behavior, global dynamics, and explicit solution forms of four distinct coupled systems of third-order rational difference equations. By employing an analytical approach based on mathematical induction and recursive substitution, we derive the general closed-form expressions for the solutions in each of the four cases under arbitrary non-zero real initial conditions. Furthermore, we provide rigorous proofs for the asymptotic behavior and the exact periodicity of these solutions. We completely characterize the structural conditions under which the trajectories exhibit periodic persistence or generate potential singularities, thereby determining the corresponding forbidden sets. These results provide a complete classification of the global behavior for these higher-order coupled systems, resolving open questions regarding their long-term dynamics and significantly extending the existing literature on non-linear rational difference equations.