Asian Journal of Probability and Statistics

This paper presents an analytical investigation of a fourth-order system of nonlinear rational difference equations, focusing on four distinct sign configurations in the denominators. For each configuration, we derive explicit closed-form expressions for the solutions, revealing a remarkable period-6 structure with distinct algebraic formulas for indices congruent to -3, -2, -1, 0, 1, 2 modulo 6. The methodology employs systematic algebraic pattern recognition combined with rigorous mathematical induction to establish the validity of these formulas for all non-negative integers p. Numerical simulations implemented in MATLAB illustrate the solution behavior for representative initial conditions, consistently showing convergence to the equilibrium point (0; 0) and providing visual confirmation of the analytical results. This work contributes to the existing literature by significantly expanding the class of exactly solvable higher-order rational difference equations, a domain where closed-form solutions remain relatively scarce. While prior studies have largely focused on lower-order or symmetric systems, the present investigation addresses an asymmetric fourth-order system with mixed delays, offering explicit formulas that highlight the intricate relationship between the index structure and the emergence of periodic patterns. The findings not only facilitate direct qualitative analysis without iterative computation but also lay the groundwork for subsequent stability investigations and generalizations. Limitations include the restriction to specific parameter choices and the absence of a rigorous stability analysis, both of which are identified as key directions for future research. Extensions to arbitrary parameters, higher-order systems, and singular cases are discussed, underscoring the broader implications of this work for discrete dynamical systems.