Abstract
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.
Similar content being viewed by others
References
G. Nicolis and I. Prigogine,Self Organization in Nonequilibrium Systems (Wiley, New York, 1977).
H. Haken,Synergetics: An Introduction (Springer, Berlin, 1978).
A. T. Winfree,The Geometry of Biological Time (Springer, New York, 1980).
A. T. Winfree,J. Theor. Biol. 16:15 (1967).
Y. Kuramoto, inInternational Symposium on Mathematical Problems in Theoretical Physics, H. Araki, ed. (Springer, New York, 1975).
Y. Aizawa,Prog. Theor. Phys. 56:703 (1976).
Y. Yamaguchi, K. Kometani, and H. Shimizu,J. Stat. Phys. 26:719 (1981).
A. H. Cohen, P. J. Holmes, and R. H. Rand,J. Math. Biol. 13:345 (1982).
Y. Yamaguchi and H. Shimizu,Physica 11D:212 (1984).
Y. Kuramoto,Prog. Theor. Physd. (Suppl.)79:223 (1984).
G. B. Ermentrout and N. Kopell,SIAM J. Math. Anal. 15:215 (1984).
N. Kopell and G. B. Ermentrout,Commun. Pure Appl. Math. 39:623 (1986).
S. Shinomoto and Y. Kuramoto,Prog. Theor. Phys. 75:1319 (1986).
H. Sakaguchi and Y. Kuramoto,Prog. Theor. Phys. 76:576 (1986).
H. Daido,Prog. Theor. Phys. 75:1460 (1986).
H. Daido,J. Phys. A 20:L629 (1987).
H. Daido,Prog. Theor. Phys. 77:622 (1987).
H. Sakaguchi, S. Shinomoto, and Y. Kuramoto,Prog. Theor. Phys. 77:1005 (1987).
H. Sakaguchi, Doctoral thesis, Kyoto University (1987) [in Japanese].
H. Daido,Phys. Rev. Lett. 61:231 (1988).
S. H. Strogatz and R. E. Mirollo,Physica D 31:143 (1988).
M. Shiino and M. Frankowicz,Phys. Lett. A 136:103 (1989).
J. Treiber and R. I. Kitney,Phys. Lett. A 134:108 (1988).
K. Satoh,J. Phys. Soc. Jpn. 58:2010 (1989).
H. E. Stanley,Introduction to Phase Transitions and Critical Phenomena (Clarendon, Oxford, 1971).
T. Okada, inLiving Things and Cooperative Phenomena, N. Saito and A. Ikegami, eds. (Tokyo University Press, Tokyo, 1976) [in Japanese].
Y. Kuramoto and I. Nishikawa,J. Stat. Phys. 49:569 (1987).
H. Daido,Prog. Theor. Phys. 81:727 (1989).
H. Daido, preprint (July 1988).
N. G. van Kampen,Can. J. Phys. 39:551 (1961); see also R. Kubo, K. Matsuo, and K. Kitahara,J. Stat. Phys. 9:51 (1973).
M. E. Fisher and M. N. Barber,Phys. Rev. Lett. 28:1516 (1972); M. Suzuki,Prog. Theor. Phys. 58:1142 (1977); M. N. Barber, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1983).
H. Daido,Prog. Theor. Phys. (Suppl.) No. 99, to appear.
R. Botet and R. Jullien,Phys. Rev. B 28:3955 (1983).
I. Nishikawa, Annual meeting of the Physical Society of Japan, September 1987, Tohoku University).
Y. Kuramoto and I. Nishikawa, inCooperative Dynamics in Complex Physical Systems, H. Takayama, ed. (Springer, Berlin, 1989).
M. Toda and R. Kudo, eds.,Statistical Physics (Iwanami, Tokyo, 1972) [in Japanese].
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Daido, H. Intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators. J Stat Phys 60, 753–800 (1990). https://doi.org/10.1007/BF01025993
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01025993