The dynamics of a system of coupled crystal oscillators (CCOST) is examined with the aim of developing a stable precision timing device. Symmetry is used to establish the existence and stability of collective patterns of oscillations in the CCOST device. We investigate N identical crystal oscillators, where each is described by a two-mode nonlinear oscillator circuit that exhibits SO(2) SO(2)-symmetry. The coupling is assumed to be identical, and two different topologies, unidirectional and bidirectional, are considered. The unidirectional topology leads to a network with SO(2) SO(2) ZN-symmetry. On the other hand, the bidirectional topology yields a network with SO(2) SO(2) DN -symmetry. The possible patterns of oscillation are classified using these symmetries and their respective isotropy subgroups. The effects of noise on the unidirectional CCOST device will be investigated with respect to phase error reduction. Phase error reduction of certain patterns will be tested against an uncoupled-averaged control group. This work will instruct on the design rules for the proposed precision timing device.
