Synchronization in networks of mutually delay-coupled phase-locked loops

Electronic components that perform tasks in a concerted way rely on a common time reference. For instance, parallel computing demands synchronous clocking of multiple cores or processors to reliably carry out joint computations. Here, we show that mutually coupled phase-locked loops (PLLs) enable synchronous clocking in large-scale systems with transmission delays. We present a phase description of coupled PLLs that includes filter kernels and delayed signal transmission. We find that transmission delays in the coupling enable the existence of stable synchronized states, while instantaneously coupled PLLs do not tend to synchronize. We show how filtering and transmission delays govern the collective frequency and the time scale of synchronization.