Symmetries and exceptional points of discrete Floquet mechanical systems*

Classical mechanical spring-mass systems with time-periodic modulation of spring stiffnesses are Floquet systems with the added feature that the solutions are real-valued and lie in a symplectic manifold. This constrains the Floquet exponents -- complex-valued generalizations of the oscillation frequencies -- to obey certain relations enabling new kinds of topologically protected phenomena and topological classifications. 

We study systems of two, three, or four oscillators coupled together via modulated springs in different geometric configurations. When phase differences among the modulations are varied, the Floquet exponents form continuous loops in the complex plane. These loops are constrained by the symmetries enjoyed by special points corresponding to in-phase and fully-out-of-phase modulation of neighboring oscillators. 

We elucidate the topological properties of these loops, and link them to robust stable collective excitations which exist even when the individual modes are unstable due to parametric amplification. Besides revealing the Floquet topological structures of mechanical systems, such finite systems with small degrees of freedom can be used as building blocks to create resonator-based driven metamaterials with desirable sound manipulation capabilities.