Floquet Theory for Second Order Linear Homogeneous Difference Equations
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.
Cite
@article{arxiv.1510.00410,
title = {Floquet Theory for Second Order Linear Homogeneous Difference Equations},
author = {Andrés M. Encinas and M. José Jiménez},
journal= {arXiv preprint arXiv:1510.00410},
year = {2015}
}