Order-reducing Form Symmetries and Semiconjugate Factorizations of Difference Equations

TL;DR

This paper explores how form symmetries and semiconjugate factorizations can reduce the order of difference equations, providing new methods for analyzing their structure and solutions, including applications to linear and exponential equations.

Contribution

It introduces a general framework for order reduction of difference equations using form symmetries and semiconjugate relations, with specific applications to linear and exponential equations.

Findings

Complete factorization of linear non-homogeneous difference equations into first-order systems.

Application of form symmetries to explain multistable behavior in exponential equations.

Reduction of higher-order difference equations through semiconjugate factorizations.

Abstract

The scalar difference equation $x_{n+1}=f_{n}(x_{n},x_{n-1},...,x_{n-k})$ may exhibit symmetries in its form that allow for reduction of order through substitution or a change of variables. Such form symmetries can be defined generally using the semiconjugate relation on a group which yields a reduction of order through the semiconjugate factorization of the difference equation of order $k+1$ into equations of lesser orders. Different classes of equations are considered including separable equations and homogeneous equations of degree 1. Applications include giving a complete factorization of the linear non-homogeneous difference equation of order $k+1$ into a system of $k+1$ first order linear non-homogeneous equations in which the coefficients are the eigenvalues of the higher order equation. Form symmetries are also used to explain the complicated multistable behavior of a separable, second order exponential equation.