On the Floquet Theory of Delay Differential Equations

TL;DR

This paper develops an analytical framework for analyzing the stability of nonlinear delay differential equations near periodic states, extending classical methods and introducing matrix continued fractions to compute Floquet eigenvalues.

Contribution

It extends Poincaré-Lindstedt and Shohat expansions to delay differential equations and introduces a systematic method for linear stability analysis using matrix continued fractions.

Findings

Approximate determination of periodic reference states.

Calculation of Floquet eigenvalues and eigensolutions.

Framework applicable to nonlinear delay differential equations.

Abstract

We present an analytical approach to deal with nonlinear delay differential equations close to instabilities of time periodic reference states. To this end we start with approximately determining such reference states by extending the Poincar'e Lindstedt and the Shohat expansions which were originally developed for ordinary differential equations. Then we systematically elaborate a linear stability analysis around a time periodic reference state. This allows to approximately calculate the Floquet eigenvalues and their corresponding eigensolutions by using matrix valued continued fractions.