Abstract
A mean-field model of nonlinearly coupled oscillators with randomly distributed frequencies and subject to independent external white noises is analyzed in the thermodynamic limit. When the frequency distribution isbimodal, new results include subcritical spontaneous stationary synchronization of the oscillators, supercritical time-periodic synchronization, bistability, and hysteretic phenomena. Bifurcating synchronized states are asymptotically constructed near bifurcation values of the coupling strength, and theirnonlinear stability properties ascertained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Scheutzow,Prob. Theory Related Fields 72:425 (1986).
A. T. Winfree,J. Theor. Biol. 16:15 (1967);The Geometry of Biological Time (Springer, New York, 1980).
L. L. Bonilla,Phys. Rev. B 35:3637 (1987), and references cited therein.
R. Mirollo and S. H. Strogatz,SUM J. Appl. Math. 50:1645 (1990).
Y. Kuramoto, inInternational Symposium on Mathematical Problems in Theoretical Physics, H. Araki, ed., Lecture Notes in Physics, Vol. 39 (Springer, New York, 1975); Y. Kuramoto and I. Nishikawa,J. Stat. Phys. 49:569 (1987).
R. Desai and R. Zwanzig,J. Stat. Phys. 19:1 (1978).
D. Dawson,J. Stat. Phys. 31:29 (1983).
L. L. Bonilla and J. M. Casado,J. Stat. Phys. 56:113 (1989).
S. H. Strogatz and R. E. Mirollo,J. Stat. Phys. 63:613 (1991).
M. Shiino,Phys. Lett. 111A:396 (1985).
L. L. Bonilla, J. M. Casado, and M. Morillo,J. Stat. Phys. 48:571 (1987); Erratum,J. Stat. Phys. 50:849 (1988).
L. L. Bonilla,Phys. Rev. Lett. 60:1398 (1988); and inFar from Equilibrium Phase Transitions, L. Garrido, ed., Lecture Notes in Physics, Vol. 319 (Springer, Berlin, 1988).
E. Lumer and B. A. Huberman,Neural Computation, submitted.
L. L. Bonilla,J. Stat. Phys. 46:659 (1987).
J. B. Keller and L. L. Bonilla,J. Stat. Phys. 42:1115 (1986).
P. C. Mathews, R. E. Mirollo, and S. H. Strogatz,Phys. Rev. Lett. 65:1701 (1990);Physica D, to appear.
H. Sakaguchi,Prag. Theor. Phys. 79:39 (1988).
R. E. Mirollo and S. H. Strogatz,J. Stat. Phys. 60:245 (1990).
G. Iooss and D. D. Joseph,Elementary Stability and Bifurcation Theory (Springer, Berlin, 1980).
S. Kogelman and J. B. Keller,SIAM J. Appl. Math. 20:619 (1971).
Y. Kuramoto,Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984), pp. 84–88.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bonilla, L.L., Neu, J.C. & Spigler, R. Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators. J Stat Phys 67, 313–330 (1992). https://doi.org/10.1007/BF01049037
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01049037