Order-Reducing Form Symmetries and Semiconjugate Factorizations of Difference Equations

The scalar difference equation xn+1 = fn(xn, xn−1, · · · , xn−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 remarkably complex, multistable behavior of a separable, second order exponential equation.