Floquet Theory for Second Order Linear Homogeneous Difference Equations

TL;DR

This paper extends Floquet's theorem to second order linear difference equations with quasi-periodic coefficients by establishing their equivalence to Chebyshev equations and deriving conditions for quasi-periodic solutions.

Contribution

It introduces a Floquet-type theorem for second order difference equations with quasi-periodic coefficients, including a method to determine the key parameter via a non-linear recurrence.

Findings

Equivalence between quasi-periodic difference equations and Chebyshev equations.

Closed-form expression for the Floquet parameter.

Necessary and sufficient condition for quasi-periodic solutions.

Abstract

In this paper we provide a version of the Floquet's theorem to be applied to any second order difference equations with quasi-periodic coefficients. To do this we extend to second order linear difference equations with quasi-periodic coefficients, the known equivalence between the Chebyshev equations and the second order linear difference equations with constant coefficients. So, any second order linear difference equations with quasi-periodic coefficients is essentially equivalent to a Chebyshev equation, whose parameter only depends on the values of the quasi-periodic coefficients and can be determined by a non-linear recurrence. Moreover, we solve this recurrence and obtaining a closed expression for this parameter. As a by-product we also obtain a Floquet's type result; that is, the necessary and sufficient condition for the equation has quasi-periodic solutions.