Solving Nonlinear Difference Equations: Insights from Three-Dimensional Systems and Numerical Examples

This paper presents a study on nonlinear difference equation systems of 6k + 3 order. The equations are of the form pn+1 =pn−(6k+2) / (±1±qn−2k rn−(4k+1) pn−(6k+2)), qn+1 =qn−(6k+2) / (±1±rn−2kpn−(4k+1) qn−(6k+2)),rn+1 =rn−(6k+2) / (±1± pn−2kqn−(4k+1) rn−(6k+2)), k ≥ 0 where n is a non-negative integer (belonging to the set N0 = N ∪ {0}) and the starting values p−l , q−l , r−l , l ∈ {0, 1, . . . , 6k+2} are arbitrary nonzero real numbers. We propose a systematic approach to solve this system, introducing a novel technique to find explicit solutions. The main outcomes of our study are the explicit solutions derived from the considered system. The study examines four different cases of this system and provides numerical examples to illustrate the results. The numerical examples demonstrate the behavior of the system for various initial conditions. The study is concluded with graphical representations of the solutions for each case, providing insights into the behavior of the systems.