On the Scaling Law of Phase Drift in Coupled Nonlinear Oscillators for Precision Timing

Abstract

Computational and experimental works reveal that the coupling of similar crystal oscillators leads to a variety of collective patterns, mainly various forms of discrete rotating waves and synchronization patterns, which have the potential for developing precision timing devices through phase drift reduction. Among all observed patterns, the standard traveling wave, in which consecutive crystals oscillate out of phase by [Formula: see text], where [Formula: see text] is the network size, leads to optimal phase drift error that scales down as [Formula: see text] as opposed to [Formula: see text] for an uncoupled ensemble. In this manuscript, we provide an analytical proof of the scaling laws, for uncoupled and coupled symmetric networks, and show that [Formula: see text] is the fundamental limit of phase-error reduction that one can obtain with a symmetric network of nonlinear oscillators of any type, not just crystals.