Abstract
We discuss ways of defining complexity in physics, and in particular for symbol sequences typically arising in autonomous dynamical systems. We stress that complexity should be distinct from randomness. This leads us to consider the difficulty of making optimal forecasts as one (but not the only) suitable measure. This difficulty is discussed in detail for two different examples: left-right symbol sequences of quadratic maps and 0–1 sequences from 1-dimensional cellular automata iterated just one single time. In spite of the seeming triviality of the latter model, we encounter there an extremely rich structure.
Preview
Unable to display preview. Download preview PDF.
References
J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985)
R. Shaw, Z. Naturforsch. 36a, 80 (1981)
T. Hogg and B.A. Huberman, Physica 22D, 376 (1986)
C.P. Bachas and B.A. Huberman, Phys. Rev. Lett. 57, 1965 (1986)
H.A. Cecatto and B.A. Huberman, Xerox preprint (1987)
S. Wolfram, Rev. Mod. Phys. 55, 601 (1983)
S. Wolfram, Commun. Math. Phys. 96, 15 (1984)
P. Grassberger, Int. J. Theoret. Phys. 25, 907 (1986)
J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages, and Computation (Addison-Wesley, New York 1979)
A.N. Kolmogorov, Three Approaches to the Quantitative Definition of Information, Probl. Inform. Transmiss. 1, 1 (1965)
G. Chaitin, J. Assoc. Comp. Mach. 13, 547 (1966)
S. Wagon, Mathem. Intell. 7, 65 (1985)
C.H. Bennett, in Emerging Syntheses in Science, D. Pines editor, 1985
S. Wolfram, Random Sequence Generation by Cellular Automata, to appear in Adv. Appl. Math.
A. Lempel and J. Ziv, IEEE Trans. Inform. Theory 22, 75 (1976)
J. Ziv and A. Lempel, IEEE Trans. Inform. Theory 23, 337 (1977); 24, 530 (1978)
T.A. Welch, Computer 17, 8 (1984)
P. Grassberger, preprint (1987), subm. to IEEE Trans. Inform. Theory
T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47, 1400 (1981)
G. Parisi, appendix in U. Frisch, Fully Developed Turbulence and Intermittency, in Proc. of Int. School on “Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics”, M. Ghil editor (North Holland, 1984)
R. Benzi et al., J. Phys. A17, 3521 (1984)
P. Grassberger, J. Stat. Phys. 45, 27 (1986)
D.R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid (Vintage Books, New York 1980)
P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Basel 1980)
D. Zambella and P. Grassberger, preprint (march 1988)
J. Dias de Deus, R. Dilao, and A. Noronha de Costa, Lisboa preprint (1984)
P. Grassberger, preprint WU-B 87-5 (1987)
F. Hofbauer, Israel J. Math. 34, 213 (1979); 38, 107 (1981); Erg. Th. & Dynam. Syst. 5, 237 (1985)
P. Collet, preprint (1986)
J.E. Hutchinson, Indiana Univ. Math. J. 30, 713 (1981)
M.F. Barnsley and S. Demko, Proc. Royal Soc. London A399, 243 (1984)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1988 Springer-Verlag
About this paper
Cite this paper
Grassberger, P. (1988). Complexity and forecasting in dynamical systems. In: Peliti, L., Vulpiani, A. (eds) Measures of Complexity. Lecture Notes in Physics, vol 314. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-50316-1_1
Download citation
DOI: https://doi.org/10.1007/3-540-50316-1_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50316-3
Online ISBN: 978-3-540-45968-2
eBook Packages: Springer Book Archive