We analyze the linear stability of the synchronized state in networks of N identical digital phase-locked loops. These are pulse-coupled oscillator arrays in which the frequency (rather than the phase) of each oscillator is updated discontinuously whenever that oscillator reaches a specific phase in its cycle. Three different coupling configurations are studied: one-way rings, two-way rings, and globally coupled arrays. In each case we obtain explicit formulas for the transient time to lock, the critical gain at which the synchronized state loses stability, and the period of the bifurcating solution at the onset of instability. Our results explain the numerical observations of de Sousa Vieira, Lichtenberg, and Lieberman.
