On the synchronization of two impulsive oscillators under communication constraints

The problem of establishing synchronization of a class of impulsive oscillators is considered. Each impulsive oscillator is modeled as a hybrid system that self resets to zero when its state reaches a threshold and is externally reset via an impulsive update law when information from other oscillators is received. At every reset to zero, the oscillators broadcast a packet and select a different communication channel. The mechanism resembles that of a firefly model in which the external resets correspond to flashes of the fireflies. Oscillators can only receive information when they are on the same channel that a packet was broadcast. This mechanism leads to the possibility of information loss. Under such a communication constraint, we show that the coupled impulsive oscillators (almost globally) synchronize their channel selections. To establish this result, we model the interconnection of oscillators as a hybrid system and apply recently developed Lyapunov stability tools. Numerical simulations are included to illustrate the results.