Abstract
The average trajectories and fluctuations around them resulting from an ensemble of noisy, nonlinear maps are analyzed. The bifurcation diagram for the average value obtained from the computer simulation of noisy maps ensemble is discussed first. Then a deterministic average equation of motion describing in an approximate way the time evolution of the average value and of the variance is analyzed numerically. This equation predicts the existence of the bifurcation gap and of the exceptional attractors for special initial points. The scaling properties of the average value and of the variance are obtained with the help of this equation.
Similar content being viewed by others
References
J. P. Crutchfield and B. A. Huberman,Phys. Lett. 77A:407 (1980).
J. P. Crutchfield, M. Nauenberg, and J. Rudnick,Phys. Rev. Lett. 46:933 (1981).
B. Shraiman, C. E. Wayne, and P. C. Martin,Phys. Rev. Lett. 46:935 (1981).
J. P. Crutchfield, J. D. Farmer, and B. A. Huberman,Phys. Rep. 92:45 (1982).
J. Heldstab, H. Thomas, T. Geisel, and G. Radons,Z. Phys. B50:141 (1983); T. Geisel, J. Heldstab, and H. Thomas,Z. Phys. B55:165 (1984).
R. Vallée, C. Delisle, and J. Chrostowski,Phys. Rev. A30:336 (1984).
G. Mayer-Kress and H. Haken,J. Stat. Phys. 26:149 (1981).
M. Feigenbaum,J. Stat. Phys. 21:669 (1979).
S. Grossman and S. Thomae,Z. Naturforsch. 32a:1353 (1977).
M. Giglio, S. Musazzi, and U. Perini,Phys. Rev. Lett. 47:243 (1981).
C. W. Smith, M. J. Tejwani, and D. A. Farris,Phys. Rev. Lett. 48:492 (1982).
R. H. Simoyi, A. Wolf, and H. L. Swinney,Phys. Rev. Lett. 49:245 (1982).
A. Libchaber, C. Laroche, and S. Fauve,J. Physique Lett. 43:L211 (1982).
P. Collet, and J.-P. Eckmann,Iterated Maps on the Interval as Dynamical Systems (Birkhäuser, Boston, 1980).
The restriction to small noise ensures that the fraction of trajectories which leave the intervalJ is negligible. In analogy with critical phenomena the expansion (4) can be expected to break down at the bifurcation points of the noiseless map but we do not concentrate on this aspect in the present paper.
T. Hogg and B. A. Huberman,Phys. Rev. A29:275 (1984).
T. Janssen and J. A. Tjon,Phys. Lett. 87A:139 (1982);J. Phys. A16:673, 697 (1983).
P. Manneville and Y. Pomeau,Phys. Lett. 75A:1 (1979).
J. E. Hirsch, B. A. Huberman, and D. J. Scalapino,Phys. Rev. A25:519 (1982).
J.-P. Eckmann,Rev. Mod. Phys. 53:643 (1981).
M. H. Jensen, P. Bak, and T. Bohr,Phys. Rev. A30:1960 (1984).
J. H. Curry, and J. A. Yorke inThe Structure of Attractors In Dynamical Systems, N. G. Markley, J. C. Martin, and W. Perozo eds., Lecture Notes in Mathematics (Springer, Berlin, 1978), Vol. 668, 48.
S. D. Feit,Comm. Math. Phys. 61:249 (1978).
G. Grebogi, E. Ott, and J. A. Yorke,Phys. Rev. Lett. 48:1507 (1982);Physica 7D:181 (1983).
Y. Gu, M. Tung, J.-M. Yuan, D. H. Feng, and M. M. Narducci,Phys. Rev. Lett. 52:701 (1984).
M. Napiórkowski,Phys. Lett. 112A:357 (1985).
F. B. Vul, Ya, G. Sinai, and K. M. Khanin,Uspekhi Mat. Nauk 39(3):3 (1984) [Russian Math. Surveys 39(3):1 (1984)].
Author information
Authors and Affiliations
Additional information
On leave from Institute for Theoretical Physics, Warsaw University, 00-681 Warsaw, Hoza 69, Poland.
Rights and permissions
About this article
Cite this article
Napiórkowski, M., Zaus, U. Average trajectories and fluctuations from noisy, nonlinear maps. J Stat Phys 43, 349–368 (1986). https://doi.org/10.1007/BF01010587
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01010587