×

Collective behavior in ensembles of globally coupled maps. (English) Zbl 0813.34045

Summary: Coherent collective behavior in an ensemble of globally coupled maps is investigated in the limit of infinite number of elements. A nonlinear Frobenius-Perron equation is derived for this system, and it is shown that it can have quasiperiodic and chaotic solutions. For the description of finite ensembles we propose a noisy nonlinear Frobenius-Perron equation and show that it gives the correct power spectrum of mean field fluctuations.

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hadley, P.; Beasley, M. R.; Wiesenfeld, K., Phys. Rev. B, 38, 8712 (1988)
[2] Wiesenfeld, K.; Bracikowski, C.; James, G.; Roy, R., Phys. Rev. Lett., 65, 1749 (1990)
[3] Strogatz, S. H.; Marcus, C. M.; Westervelt, R. M.; Mirollo, R. E., Physica D, 36, 23 (1989)
[4] Sompolinsky, H.; Golomb, D.; Kleinfeld, D., Phys. Rev. A, 43, 6990 (1991)
[5] Golomb, D.; Hansel, D.; Shraiman, B.; Sompolinsky, H., Phys. Rev. A, 45, 3516 (1992)
[6] Nichols, S.; Wiesenfeld, K., Phys. Rev. A, 45, 8430 (1992)
[7] Strogatz, S. H.; Mirollo, R. E., Phys. Rev. E, 47, 220 (1993)
[8] Wiesenfeld, K.; Hadley, P., Phys. Rev. Lett., 62, 1335 (1989)
[9] Hakim, V.; Rappel, W.-J., Phys. Rev. A, 46, R7347 (1992)
[10] Nakagawa, N.; Kuramoto, Y., Prog. Theor. Phys., 89, 313 (1993)
[11] Kaneko, K., Phys. Rev. Lett., 63, 219 (1989)
[12] Kaneko, K., Physica D, 41, 137 (1990)
[13] Kaneko, K., Physica D, 55, 368 (1992)
[14] Fabiny, L.; Wiesenfeld, K., Phys. Rev. A, 43, 2640 (1991)
[15] Kometani, K.; Shimizu, H., J. Stat. Phys., 13, 473 (1975)
[16] Desai, R. C.; Zwanzig, R., J. Stat. Phys., 19, 1 (1978)
[17] Bonilla, L. L.; Casado, J. M.; Morillo, M., J. Stat. Phys., 48, 571 (1987)
[18] Bonilla, L. L.; Neu, J. C.; Spigler, R., J. Stat. Phys., 67, 313 (1992)
[19] Dawson, D. A., J. Stat. Phys., 31, 29 (1983)
[20] Kaneko, K., Phys. Rev. Lett., 65, 1391 (1990)
[21] Perez, G.; Sinha, S.; Cerdeira, H., Phys. Rev. A, 45, 5469 (1992)
[22] Griniasty, M.; Hakim, V., Phys. Rev. E (1993), submitted
[23] Lasota, A.; Mackay, M. C., Probabilistic Properties of Deterministic Systems (1985), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
[24] Ershov, S. V., Phys. Lett. A, 177, 180 (1993)
[25] Choi, M. Y.; Huberman, B. A., Phys. Rev. A, 28, 1204 (1983)
[26] Jakobson, M. V., Commun. Math. Phys., 81, 39 (1981)
[27] Perez, G.; Cerdeira, H. A., Phys. Rev. E, 49, R15 (1994)
[28] Grassberger, P.; Schreiber, T.; Schaffrath, C., Int. J. Bifurc. Chaos, 1, 521 (1991)
[29] E.J. Kostelich and T. Schreider, Phys. Rev. E 3 (48) 1752.; E.J. Kostelich and T. Schreider, Phys. Rev. E 3 (48) 1752.
[30] Sinha, S., Phys. Rev. Lett., 69, 3306 (1992)
[31] Ding, M.; Wille, L. T., Phys. Rev. E, 48, R1605 (1993)
[32] Pikovsky, A. S., Phys. Rev. Lett., 71, 653 (1993)
[33] Wayland, R.; Bromley, D.; Pickett, D.; Passamante, A., Phys. Rev. Lett., 70, 580 (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.