Journal of Statistical Physics

, Volume 49, Issue 3, pp 569–605

Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities

  • Yoshiki Kuramoto
  • Ikuko Nishikawa

DOI: 10.1007/BF01009349

Cite this article as:
Kuramoto, Y. & Nishikawa, I. J Stat Phys (1987) 49: 569. doi:10.1007/BF01009349


A model dynamical system with a great many degrees of freedom is proposed for which the critical condition for the onset of collective oscillations, the evolution of a suitably defined order parameter, and its fluctuations around steady states can be studied analytically. This is a rotator model appropriate for a large population of limit cycle oscillators. It is assumed that the natural frequencies of the oscillators are distributed and that each oscillator interacts with all the others uniformly. An exact self-consistent equation for the stationary amplitude of the collective oscillation is derived and is extended to a dynamical form. This dynamical extension is carried out near the transition point where the characteristic time scales of the order parameter and of the individual oscillators become well separated from each other. The macroscopic evolution equation thus obtained generally involves a fluctuating term whose irregular temporal variation comes from a deterministic torus motion of a subpopulation. The analysis of this equation reveals order parameter behavior qualitatively different from that in thermodynamic phase transitions, especially in that the critical fluctuations in the present system are extremely small.

Key words

Large dissipative systempopulation of limit cycle oscillatorsorder parameterphase transition via mutual entrainmentapproximate invariant measuredynamical extension of self-consistent equationcritical slowing downanomalous critical fluctuation

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • Yoshiki Kuramoto
    • 1
  • Ikuko Nishikawa
    • 1
  1. 1.Department of PhysicsKyoto UniversityKyotoJapan