Dynamics and Expressions of Solutions of Nonlinear Difference Equations X_(n+1)=(x_(n-3) X_(n-6))/(±x_(n-2)±x_(n-2) X_(n-3) X_(n-6) )

Abstract

Nonlinear difference equations provide a framework for modeling natural phenomena in nonlinear sciences. In this paper, we investigate the periodicity, boundedness, oscillation, stability, and exact solutions of such equations. Employing the standard iteration method, we derive closed-form solutions and analyze the stability of equilibrium points using established theorems. Numerical simulations, implemented in Wolfram Mathematica, corroborate the theoretical findings. The proposed method can be readily extended to other rational recursive problems. This paper investigates the dynamical behavior of solutions to the rational difference equation x_(n+1)=(x_(n-3) x_(n-6))/(±x_(n-2)±x_(n-2) x_(n-3) x_(n-6) ) where the initial conditions are arbitrary nonzero real numbers. We analyze the stability properties, periodic solutions, and long-term behavior of this equation, employing both analytical and numerical approaches to characterize its dynamics.