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arXiv · 0804.3579

Order-reducing Form Symmetries and Semiconjugate Factorizations of Difference Equations

H. Sedaghat
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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.

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H. Sedaghat. 2008-05-27. Order-reducing Form Symmetries and Semiconjugate Factorizations of Difference Equations. https://arxiv.org/abs/0804.3579

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