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The collective behavior in a population of globally coupled oscillators withrandomly distributed frequencies is studied when the natural frequencydistribution does not possess an even symmetry with respect to the averagenatural frequency of oscillators. We study the special case of absence ofsymmetry induced by a group of scaling transformations of the continuousdistribution of frequencies. When coupling between oscillators is increasedbeyond a critical threshold favoring spontaneous synchronization, we found thatthe variation in the velocity of the traveling wave depends on the extent ofasymmetry in the natural frequency distribution. In particular for largecoupling this velocity is the average natural frequency whereas at the onset ofsynchronization it corresponds to the frequency where the Hilbert Transform ofthe frequency distribution vanishes.

Synchronization of globally coupled oscillators without symmetry in the distribution of natural frequencies