Group reduction, formulas solutions and asymptotic behavior of a class of fourth order difference equations

The symmetry method is a powerful and systematic approach for solving difference equations. It leverages the concept of transformations that leave a difference equation invariant, simplifying its structure and often reducing the equation to a solvable form. In this paper, the symmetry method is employed to study some class of difference equations. Using analytical techniques and computational tools, we derive explicit solutions for these equations and establish conditions for the existence of periodic solutions. Stability analysis is performed to identify non-hyperbolic points. Furthermore, some asymptotic properties of the difference equations are explored, with results and graphs illustrating how initial conditions and parameter values influence the behavior of the solutions.