Closed-Form Solutions and Periodic Dynamics of a Four-Dimensional Eighth-Order System of Rational Difference Equations

Abstract

This paper investigates the following eighth-order nonlinear system of rational difference equations for different cases:
\begin{equation*}
\left\{
\begin{array}{l}
P_{n+1} =\frac{P_{n-7}}{\pm1\pm Q_{n-1}R_{n-3}S_{n-5}P_{n-7}} \\[10pt] 
Q_{n+1} =\frac{Q_{n-7}}{\pm1\pm R_{n-1}S_{n-3}P_{n-5}Q_{n-7}} \\[10pt] 
R_{n+1} =\frac{R_{n-7}}{\pm1\pm S_{n-1}P_{n-3}Q_{n-5}R_{n-7}} \\[10pt] 
S_{n+1} =\frac{S_{n-7}}{\pm1\pm P_{n-1}Q_{n-3}R_{n-5}S_{n-7}} 

\end{array}
\right.
\end{equation*}
where $n$ is a non-negative integer and the initial conditions $P_{-7},..., P_{0}, Q_{-7},..., Q_{0}, R_{-7},..., R_{0}, \\ S_{-7},..., S_{0} $ are arbitrary real numbers. The study derives explicit closed-form solutions under specific conditions. These solutions facilitate rigorous analysis and provide an exact description of the long-term behavior of the system. Finally, numerical simulations are presented to illustrate key dynamical properties, including boundedness, periodicity, and sensitivity to initial conditions.