J. R. Pierce has recently proposed a system for synchronizing an arbitrary number of geographically separated oscillators, and, under the assumption of zero transmission delays between stations, has shown that a certain linear model of the system is stable in the sense that all of the station frequencies approach a common final value as t → ∞.
This paper reports on some results concerning the dynamic behavior of Pierce's linear model. The results take into account transmission delays. More explicitly, we prove that if a certain set of simple inequalities involving the delays is satisfied, then the system is stable and the oscillator frequencies approach their common final value at an exponential rate. These inequalities have the property that they are always satisfied for sufficiently small delays. This paper presents a simple example showing that the system can be unstable when the inequalities are not met. In addition, we present some information concerning the rate of decay of the natural modes of stable systems, discuss an alternative stability criterion not involving the transmission delays and derive an explicit expression for the final frequency. Finally, we discuss the mathematical relationship between Pierce's model and earlier models of synchronization systems.