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Being driven by the goal of finding edge modes and of explaining the occurrence of edge modes in the case of time-modulated metamaterials in the high-contrast and subwavelength regime, we analyse the topological properties of Floquet normal forms of periodically parameterized time-periodic linear ordinary differential equations $$\left\{ \frac{d}{dt}X = A_\alpha (t)X\right\} _{\alpha \in {\mathbb {T}}^d}$$ d dt X = A α ( t ) X α ∈ T d . In fact, our main goal being the question whether an analogous principle as the bulk-boundary correspondence of solid-state physics is possible in the case of Floquet metamaterials, i.e., subwavelength high-contrast time-modulated metamaterials. This paper is a first step in that direction. Since the bulk-boundary correspondence states that topological properties of the bulk materials characterize the occurrence of edge modes, we dedicate this paper to the topological analysis of subwavelength solutions in Floquet metamaterials. This work should thus be considered as a basis for further investigation on whether topological properties of the bulk materials are linked to the occurrence of edge modes. The subwavelength solutions being described by a periodically parameterized time-periodic linear ordinary differential equation $$\left\{ \frac{d}{dt}X = A_\alpha (t)X\right\} _{\alpha \in {\mathbb {T}}^d}$$ d dt X = A α ( t ) X α ∈ T d , we put ourselves in the general setting of periodically parameterized time-periodic linear ordinary differential equations and introduce a way to (topologically) classify a Floquet normal form F,  P of the associated fundamental solution $$\left\{ X_\alpha (t) = P(\alpha ,t)\exp (tF_\alpha )\right\} _{\alpha \in {\mathbb {T}}^d}$$ X α ( t ) = P ( α , t ) exp ( t F α ) α ∈ T d . This is achieved by analysing the topological properties of the eigenvalues and eigenvectors of the monodromy matrix $$X_\alpha (T)$$ X α ( T ) and the Lyapunov transformation $$P(\alpha ,t)$$ P ( α , t ) . The corresponding topological invariants can then be applied to the setting of Floquet metamaterials. In this paper these general results are considered in the case of a hexagonal structure. We provide two interesting examples of topologically non-trivial time-modulated hexagonal structures.

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.