Exact Representation Formulas for a Triple Intertwined Periodic Recurrence System with Hyperbolic-Tangent Coupling

This paper presents a new analytical framework for studying a class of three-dimensional symmetric systems within the theory of difference equations, constituting a natural extension of structurally reducible models in one and two dimensions. Despite the classical focus on linear equations or low-dimensional systems, the problem of solving nonlinear systems with intertwined structures and interdependent delays remains largely unexplored. Drawing on recent structural developments, we define a periodic symmetric system based on three sequences interacting through fractional hyperbolic tangent-type forms and show how this structure reveals embedded linear recurrences governing the internal evolution. By exploiting symmetry and periodicity properties, we derive exact representation formulas and highlight the structural mechanisms that enable a deeper understanding of the system’s dynamic behavior, despite its nonlinear nature and the complex interlacing of its indices. This study thus contributes to the expansion of the theory of solvable nonlinear systems and provides a unified approach to high- dimensional symmetric structures. To illustrate the practical relevance of the proposed framework, a cyclic SIR-type interaction interpretation is presented, supported by numerical simulations that demonstrate the impact of parameter symmetry on the system’s dynamical behavior.