Sufficient Conditions for Fast Switching Synchronization in Time-Varying Network Topologies

In previous work [J. D. Skufca and E. Bollt, Mathematical Biosciences and Engineering, 1 (2004), pp. 347-359], empirical evidence indicated that a time-varying network could propagate sufficient information to allow synchronization of the sometimes coupled oscillators, despite an instantaneously disconnected topology. We prove here that if the network of oscillators synchronizes for the static time-average of the topology, then the network will synchronize with the time-varying topology if the time-average is achieved sufficiently fast. Fast switching, fast on the time-scale of the coupled oscillators, overcomes the desynchronizing decoherence suggested by disconnected instantaneous networks. This result agrees in spirit with that of [J. D. Skufca and E. Bollt, Mathematical Biosciences and Engineering, 1 (2004), pp. 347-359] where empirical evidence suggested that a moving averaged graph Laplacian could be used in the master-stability function analysis [L. M. Pecora and T. L. Carroll, Phys. Rev. Lett., 80 (1998), pp. 2109-2112]. A new fast switching stability criterion herein gives sufficiency of a fast switching network leading to synchronization. Although this sufficient condition appears to be very conservative, it provides new insights about the requirements for synchronization when the network topology is time-varying. In particular, it can be shown that networks of oscillators can synchronize even if at every point in time the frozen-time network topology is insufficiently connected to achieve synchronization.