Onset of Collective Rhythms in Large Populations of Coupled Oscillators

The kind of complex systems of our present concern is large populations of coupled limit-cycle oscillators with frequency distribution. Such systems are by no means an invention motivated simply by mathematical curiosity. Quite on the contrary, their study would be of considerable practical value, for the same kind of systems are not rare in the real world as we realize when looking into living organisms.1) Yet we will presently be interested not so much in biological applications as in unique dynamical features shared commonly by random populations of coupled oscillators in general. Among others, we will focus on ‘frequency condensation’ by which we mean that the natural frequencies with continuous distribution change with the strength of mutual coupling until eventually a finite fraction of the population comes to share a common frequency. This remarkable behavior could be the origin of collective oscillation i.e. the oscillation of the whole population like a single giant oscillator. The same behavior would make it possible for synchronizing waves to propagate with surprizing robustness over arbitrarily long distances, which could be important in information processing in living systems.