Consider a distributed network on a finite simple graph @math with diameter @math and maximum degree @math , where each node has a phase oscillator revolving on @math with unit speed. Pulse-coupling is a class of distributed time evolution rule for such networked phase oscillators inspired by biological oscillators, which depends only upon event-triggered local pulse communications. In this paper, we propose a novel inhibitory pulse-coupling and prove that arbitrary phase configuration on @math synchronizes by time @math if @math is a tree and @math . We extend this pulse-coupling by letting each oscillator throttle the input according to an auxiliary state variable. We show that the resulting adaptive pulse-coupling synchronizes arbitrary initial configuration on @math by time @math if @math is a tree. As an application, we obtain a universal randomized distributed clock synchronization algorithm, which uses @math memory per node and converges on any @math with expected worst case running time of @math .
The coordinated motion of multi-agent systems and oscillator synchronization are two important examples of networked control systems. In this technical note, we consider what effect multiple, non-commensurate (heterogeneous) communication delays can have on the consensus properties of large-scale multi-agent systems endowed with nonlinear dynamics. We show that the structure of the delayed dynamics allows functionality to be retained for arbitrary communication delays, even for switching topologies under certain connectivity conditions. The results are extended to the related problem of oscillator synchronization.