Method of Lie Symmetry for analytical solutions, periodicity and attractivity of a family of tenth-order difference equations

TL;DR

This paper applies Lie symmetry methods to analyze a family of tenth-order difference equations, deriving analytical solutions, exploring periodicity, and studying their long-term behavior.

Contribution

It introduces a generalized approach using Lie symmetries to solve and analyze high-order difference equations, extending existing results.

Findings

Derived new analytical solutions for the difference equations

Established conditions for periodicity of solutions

Analyzed attractivity and long-term behavior of solutions

Abstract

Symmetry is a powerful tool for finding analytical solutions to differential equations, both partial and ordinary, via the similarity variables or via the invariance of the equation under group transformations. It is the largest group of transformations that leaves the differential equation invariant. It is now known that this differential equation method plays the same role when it comes to the study of difference equations. Difference equations can be used to model various phenomena where the changes occur in discrete manner. The use of symmetries on recurrence equations, usually, leads to reductions of order and hence ease the process of finding their solutions. One of the aims of this work is to employ symmetries to generalize some results in the literature. We present new generalized formula solutions of a class of difference equations and we investigate the periodicity and behavior of theses solutions.