Floquet Theory for Linear Time-Periodic Delay Differential Equations Using Orthonormal History Functions

In the usual approach to determining the stability of a time-periodic delay differential equation (DDE), the DDE is converted into an approximate system of time-periodic ordinary differential equations (ODEs) using Galerkin approximations. Later, Floquet theory is applied to these ODEs. Alternatively, semi-discretization-like approaches can be used to construct an approximate Floquet transition matrix (FTM) for a DDE. In this paper, we develop a method to obtain the FTM directly. Our approach is analogous to the Floquet theory for ODEs: we consider one polynomial basis function at a time as the history function and stack the coefficients of the corresponding DDE solutions to construct the FTM. The largest magnitude eigenvalue of the FTM determines the stability of the DDE. Since the obtained FTM is an approximation of the actual infinite-dimensional FTM, the criterion developed for stability is approximate. We demonstrate the correctness and efficacy of our method by studying several candidate DDEs with time-periodic parameters and comparing the results with those obtained from the Galerkin approximations.