Improving the Frequency Precision of Oscillators by Synchronization

Improving the frequency precision by synchronizing a lattice of oscillators is studied in the phase reduction limit. For the most commonly studied case of purely dissipative phase coupling (the Kuramoto model) I confirm that the frequency precision of N oscillators perturbed by independent noise sources is improved by a factor N as expected from simple averaging arguments. In the presence of reactive coupling, such as will typically be the case for non-dissipatively coupled oscillators based on high-Q resonators, the synchronized state consists of target like waves radiating from a local source which is a region of higher frequency oscillators. In this state all the oscillators evolve with the same frequency, however I show that the improvement of the frequency precision is independent of N for large N, but instead depends on the disorder and reflects the dependence of the frequency of the synchronized state on just those oscillators in the source region of the waves.