Asymptotic Floquet theory for first order ODEs with finite Fourier series perturbation and its applications to Floquet metamaterials

TL;DR

This paper develops an asymptotic theory for Floquet exponents in linear differential systems with periodic coefficients, and applies it to analyze Floquet metamaterials and exceptional points in resonator systems.

Contribution

It introduces a full asymptotic expansion for Floquet exponents and characterizes asymptotic exceptional points in Floquet metamaterials.

Findings

Constant order Floquet exponents with higher multiplicity are linearly perturbed.

Asymptotic exceptional points occur under specific frequency ratios related to system geometry.

The theory applies to systems with analytically dependent periodic coefficients.

Abstract

Our aim in this paper is twofold. Firstly, we develop a new asymptotic theory for Floquet exponents. We consider a linear system of differential equations with a time-periodic coefficient matrix. Assuming that the coefficient matrix depends analytically on a small parameter, we derive a full asymptotic expansion of its Floquet exponents. Based on this, we prove that only the constant order Floquet exponents of multiplicity higher than one will be perturbed linearly. The required multiplicity can be achieved via folding of the system through certain choices of the periodicity of the coefficient matrix. Secondly, we apply such an asymptotic theory for the analysis of Floquet metamaterials. We provide a characterization of asymptotic exceptional points for a pair of subwavelength resonators with time-dependent material parameters. We prove that asymptotic exceptional points are obtained if the frequency components of the perturbations fulfill a certain ratio, which is determined by the geometry of the dimer of subwavelength resonators.