Abstract
The cumulants of the information loss are discussed as characteristic measures of dynamical chaos. They are extensions of the Liapunov exponent and Kolmogorov entropy, which are given by mean values of the information loss. The most important cumulant of higher than first order is the variance. It is discussed in particular for the logistic map.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Bai-Lin,Chaos (World Scientific, Singapore, 1985).
H. G. Schuster,Deterministic Chaos (Physik-Verlag, Weinheim, Federal Republic of Germany, 1984).
H. Haken (ed.),Chaos and Order in Nature (Springer, Berlin, 1981).
H. Haken (ed.),Evolution of Order and Chaos (Springer, Berlin, 1982).
H. Haken,Advanced Synergetics (Springer, Berlin, 1984).
R. H. G. Hellemann and G. Ioos (eds.),Les Houches Summerschool on “Chaotic Behaviour in Deterministic Systems” (North-Holland, Amsterdam, 1983).
R. M. May,Nature 261:459 (1976).
S. Grossmann and S. Thomae,Z. Naturforsch. 32A:1353 (1977).
P. J. Myrberg,Ann. Acad. Sci. Fenn. A 256 (1958);268 (1959);326/3 (1963).
H. O. Peitgen and P. H. Richter,Harmonie und Kosmos; andMorphologie komplexer Grenzen (Forschungsschwerpunkt Dynamische Systeme, Universität Bremen, 1984).
V. I. Oseledec,Trans. Moscow Math. Soc. 19:197 (1968).
G. Benettin, L. Galgani, and J. Strelcyn,Phys. Rev. A 14:2338 (1976).
D. Ruelle,N. Y. Acad. Sci. 136:229 (1981).
O. E. Rössler, inNonlinear Phenomena in Chemical Dynamics, C. Vidal and A. Pacault, eds. (Springer-Verlag, 1981).
P. Grassberger and I. Procaccia,Phys. Rev. Lett. 50:346 (1983).
P. Grassberger and I. Procaccia,Physica 9D:189 (1983);13D:34 (1984).
P. Grassberger, Estimating the fractional dimensions and entropies of strange attractors, Preprint, University of Wuppertal (1984) (to appear inChaos—An Introduction, A. V. Holden, ed.).
A. Rényi,Probability Theory (North-Holland, Amsterdam, 1970).
A. N. Kolmogorov,Dokl. Akad. Nauk SSSR 98:527 (1959);124:754 (1959).
Ya. G. Sinai,Dokl. Akad. Nauk SSSR 124:768 (1959).
J. D. Farmer,Z. Naturforsch. 37a:1304 (1982).
R. Shaw,Z. Naturforsch. 36a:80 (1981).
Y. Termonia,Phys. Rev. A 29:1612 (1982).
D. Ruelle, International Conference on Bifurcation Theory, November 1977, New York.
D. Ruelle, inProceedings International Mathematical Physics Conference (Rome, 1977).
W. Gröbner and N. Hofreiter,Integraltafel II (Springer-Verlag, New York, 1966).
I. S. Gradshteyn and I. M. Ryzhik,Tables of Integrals, Series and Products (Academic Press, New York, 1980).
G. M. Fikhtengol'ts,Kurs Differential'nogo i integral'nogo ischisleniay (Moscow, 1947–1949), Vol. II, pp. 614, 643.
P. Collet and J. P. Eckmann,Iterated Maps of the Interval As Dynamical Systems (Birkhäuser, Boston, 1980).
K.-P. Dörpelkus, private communication.
M. J. Feigenbaum,J. Stat. Phys. 19:25 (1978).
M. J. Feigenbaum,Universal Behaviour in Nonlinear Systems (Los Alamos Science, 1980).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schlögl, F. The variance of information loss as a characteristic quantity of dynamical chaos. J Stat Phys 46, 135–146 (1987). https://doi.org/10.1007/BF01010336
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01010336