Intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators

A theory of intrinsic fluctuations is developed of a phase ordering parameter for large populations of weakly and uniformly coupled limit-cycle oscillators with distributed native frequencies. In particular it is shown that the intensity as well as the correlation time of fluctuations exhibit power-law divergence at the onset of mutual entrainment with critical exponents which depend on whether the coupling strength approaches the threshold from below or above. This peculiar feature is demonstrated by numerical simulations mainly through finite-size scaling analyses. In the course of exploring its origin, we encounter a new concept termed a “correlation frequency” which provides a natural interpretation of the finite-size scaling laws. A comment is given on a recent theory by Kuramoto and Nishikawa to clarify why it contradicts our results.

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