Closed-Form Solutions of a Nonlinear Rational Second-Order Three-Dimensional System of Difference Equations

In this paper, we investigate the behavior of solutions to a nonlinear system of rational difference equations of order two, defined by xn+1=xnyn−1yn(a+bxnyn−1),yn+1=ynzn−1zn(c+dynzn−1),zn+1=znxn−1xn(e+fznxn−1), where n denotes a nonzero integer; the parameters a,b,c,d,e,f are real constants; and the initial conditions x−1,x0,y−1,y0,z−1,z0 are nonzero real numbers. By applying a suitable variable transformation, we reduce the original coupled system to three independent rational difference equations. This allows for separate analysis using established methods for second-order nonlinear recurrence relations. We derive explicit solutions and examine the qualitative behavior, including boundedness and periodicity, under different conditions. Our findings contribute to the theory of rational difference equations and offer insights for higher-order systems in applied sciences.