  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.
