Order-reducing Form Symmetries and Semiconjugate Factorizations of Difference Equations
Abstract
Description
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.
25 pages, 2 figures; reduction of order based on the new concept of form symmetry and semiconjugate factorization; Version 2 adds a converse to Theorem 2, a new example and several remarks; it also updates references (two new papers on this topic accepted) and corrects minor errors
25 pages, 2 figures; reduction of order based on the new concept of form symmetry and semiconjugate factorization; Version 2 adds a converse to Theorem 2, a new example and several remarks; it also updates references (two new papers on this topic accepted) and corrects minor errors