Abstract
The nature of a very high-dimensional chaotic attractor in an infinite-dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of high-dimensional attractor in the phase space. We introduce two complementary bases from which the attractor is observed, one the Lyapunov basis composed of the Lyapunov vectors and the another the Fourier basis composed of the Fourier modes. We introduce the “exterior” subspaces on the basis of the Lyapunov vectors and observe the chaotic motion projected onto these exteriors. It is shown that a certain statistical property of the projected motion changes markedly as the exterior subspace “goes out” of the attractor. The origin of such a phenomenon is attributed to more fundamental features of our attractor, which become manifest when the attractor is observed from the Lyapunov basis. A counterpart of the phenomenon can be observed also on the Fourier basis because there is a statistical one-to-one correspondence between the Lyapunov vectors and the Fourier modes. In particular, a statistical property of the high-pass filtered time series reflects clearly the difference between the interior and the exterior of the attractor.
Similar content being viewed by others
References
A. S. Monin,Sov. Phys. Usp. 21:429 (1978); J. P. Eckmann,Rev. Mod. Phys. 53:643 (1981); R. H. G. Helleman, in W. Horton, L. Reichl, and V. Szebehely, eds.,Long-time Prediction in Dynamics, p. 95 (John Wiley & Sons, New York, 1982).
E. N. Lorenz,J. Atmos. Sci. 20:130 (1963); J. B. McLaughlin and P. C. Martin,Phys. Rev. A 12:186 (1975); H. Yahata, in Y. Kuramoto, ed.,Chaos and Statistical Methods, p. 232 (Springer, Berlin, 1984).
A. S. Monin and A. M. Yaglom,Statistical Fluid Mechanics (MIT Press, Cambridge, 1975); T. Tatsumi,Adv. Appl. Phys. 20:39 (1980).
J. D. Farmer,Physica 4D:336 (1982).
Y. Pomeau, A. Pumir, and P. Pelce,J. Stat. Phys. 37:39 (1984); P. Manneville, A. Pumir, and L. Tuckerman, preprint.
K. Ikeda,Opt. Commun. 30:257 (1979); K. Ikeda, H. Daido, and O. Akimoto,Phys. Rev. Lett. 45:709 (1980); K. Ikeda and O. Akimoto, in Y. Kuramoto, ed.,Chaos and Statistical Methods, p. 249 (Springer, Berlin, 1984).
K. Ikeda and O. Akimoto, in L. Mandel and E. Wolf, eds.,Coherence and Quantum Optics V, p. 701 (Plenum, New York, 1984).
V. I. Oseledec,Trans. Moscow Math. Soc. 19:197 (1968); I. Shimada and T. Nagashima,Prog. Theor. Phys. 61:1605 (1979).
J. Kaplan and J. Yorke, in H. O. Pietgen and H. O. Walther, eds.,Functional Differential Equations and Approximations of Fixed Points (Springer, Berlin/New York, 1979).
R. H. Kraichnan,Phys. Fluids 10:2080 (1967).
U. Frisch and R. Morf,Phys. Rev. A 23:2673 (1981).
A. Y. Kuo and S. Corrsin,J. Fluid Mech. 50:285 (1971).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ikeda, K., Matsumoto, K. Study of a high-dimensional chaotic attractor. J Stat Phys 44, 955–983 (1986). https://doi.org/10.1007/BF01011917
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01011917