Abstract
We analyze a large system of limit-cycle oscillators with mean-field coupling and randomly distributed natural frequencies. We prove that when the coupling is sufficiently strong and the distribution of frequencies has sufficiently large variance, the system undergoes “amplitude death”-the oscillators pull each other off their limit cycles and into the origin, which in this case is astable equilibrium point for the coupled system. We determine the region in couplingvariance space for which amplitude death is stable, and present the first proof that the infinite system provides an accurate picture of amplitude death in the large but finite system.
Similar content being viewed by others
References
S. S. Wang and H. G. Winful,Appl. Phys. Lett. 52:1774 (1988);53:1894 (1988).
P. Hadley, M. R. Beasley, and K. Wiesenfeld,Phys. Rev. B 38:8712 (1988);Appl. Phys. Lett. 52:1619 (1988).
J. Benford, H. Sze, W. Woo, R. R. Smith, and B. Harteneck,Phys. Rev. Lett. 62:969 (1989).
A. T. Winfree,J. Theor. Biol. 16:15 (1967).
A. T. Winfree,The Geometry of Biological Time (Springer-Verlag, New York, 1980);When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias (Princeton University Press, Princeton, New Jersey, 1987).
L. Glass and M. C. Mackey,From Clocks to Chaos: The Rhythms of Life (Princeton University Press, Princeton, New Jersey, 1988).
J. Grasman and M. J. W. Jansen,J. Math. Biol. 7:171 (1979).
D. C. Michaels, E. P. Matyas, and J. Jalife,Circ. Res. 61:704 (1987); J. Jalife,J. Physiol. 356:221 (1984).
T. Pavlidis,Biological Oscillators: Their Mathematical Analysis (Academic Press, New York, 1973).
A. H. Cohen, P. J. Holmes, and R. H. Rand,J. Math. Biol. 13:345 (1982).
N. Kopell, inNeural Control of Rhythmic Movement in Vertebrates, A. H. Cohen, S. Rossignol, and S. Grillner, eds. (Wiley, New York, 1988); N. Kopell and G. B. Ermentrout,Commun. Pure Appl. Math. 39:623 (1986).
G. B. Ermentrout and N. Kopell,SIAM J. Math. Anal. 15:215 (1984).
A. Sherman, J. Rinzel, and J. Keizer,Biophys. J. 54:411 (1988).
Y. Kuramoto and I. Nishikawa,J. Stat. Phys. 49:569 (1987).
H. Sakaguchi, S. Shinomoto, and Y. Kuramoto,Prog. Theor. Phys. 77:1005 (1987);79:1069 (1988).
H. Daido,Phys. Rev. Lett. 61:231 (1988).
S. H. Strogatz and R. E. Mirollo,J. Phys. A 21:L699 (1988);Physica D 31:143 (1988).
Y. Aizawa,Prog. Theor. Phys. 56:703 (1976).
Y. Yamaguchi and H. Shimizu,Physica D 11:212 (1984).
M. Ohsuga, Y. Yamaguchi, and H. Shimizu,Biol. Cybernet. 51:325 (1985).
K. Bar-Eli,Physica D 14:242 (1985).
G. B. Ermentrout, inNonlinear Oscillations in Biology and Chemistry, H. Othmer, ed. (Springer, New York, 1986).
G. B. Ermentrout and W. C. Troy,SIAM J. Appl. Math. 46:359 (1986).
D. Aronson, G. B. Ermentrout, and N. Kopell,Physica D, in press.
G. B. Ermentrout,Physica D, in press.
M. Shiino and M. Frankowicz,Phys. Lett. A 136:103 (1989).
H. Sakuguchi,Prog. Theor. Phys. 80:743 (1988).
L. L. Bonilla, J. M. Casado, and M. Morillo,J. Stat. Phys. 48:571 (1987); L. L. Bonilla,Phys. Rev. Lett. 60:1398 (1988).
M. Shiino,Phys. Lett. A 111:396 (1985).
G. B. Ermentrout and N. Kopell,SIAM J. Appl. Math. 50:125 (1990).
H. Haken,Synergetics (Springer, Berlin, 1978).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mirollo, R.E., Strogatz, S.H. Amplitude death in an array of limit-cycle oscillators. J Stat Phys 60, 245–262 (1990). https://doi.org/10.1007/BF01013676
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01013676