Abstract
The discovery of erratic behaviour in deterministic systems opened entirely new perÂspectives in the comprehension of dynamical behaviour of nonlinear systems. The existence of broad-band spectra no longer necessarily requires a coupling with an external uncontrollable thermal bath. Chaos, providing an information flow from irrelevant to relevant digits, naturally transforms the indetermination on the initial condition into a seemingly stochastic behaviour in time domain [1]. New classes of indicators have been consequently introduced, which allow to distinguish between truly stochastic motion and low-dimensional chaotic behaviour: Lyapunov exponents, metric entropy and fractal dimensions are dynamical invariants which measure the degree of chaoticity [2].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.D. Farmer, Z. Naturforsch. 37A, 1304 (1983)
J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985); R. Badii and A. Politi, in Instabilities and Chaos in Quantum Optics II, N.B. Abraham, F.T. Arecchi, and L.A. Lugiato eds. ( Plenum, New York, 1988 ), p. 335
K. Kaneko, Prog. Theor. Phys. 72, 480 (1984)
Y. Pomeau, A. Pumir, and P. Pelce, J. Stat. Phys. 37, 39, (1984); J,-P. Eckmann and C. E. Wayne, J. Stat. Phys. 50, 853 (1988)
F. Ledrappier, Commun. Math. Phys. 81, 229 (1981)
R. Badii and A. Politi in Fractals in Physics, L. Pietronero and E. Tosatti eds., ( Elsevier, Amsterdam 1986 ), p. 453
A. Torcini, G. D’Alessandro, and A.Politi, in preparation
P. Grassberger, “Information content and predictability of lumped and distributed dynamical systems”, preprint, Wuppertal WB 87–8
G. Mayer-Kress and K. Kaneko, J. Stat. Phys. 54, 1489 (1989)
L.A. Bunimovich and Ya G. Sinai, Nonlinearity 1, 491 (1989)
P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983)
R. Badii and A. Politi, Phys. Rev. Lett. 52, 1661 (1984)
E. Ott., W. Withers, and J.A. Yorke, J. Stat. Phys. 36, 687 (1984)
G. Mayer-Kress, in Directions in Chaos, Hao Bai-Lin ed. ( World Scientific, Singapore, 1987 )
J.-P. Eckmann and D. Ruelle, “Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems”, preprint
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Plenum Press, New York
About this chapter
Cite this chapter
Politi, A., D’Alessandro, G., Torcini, A. (1989). Fractal Dimensions in Coupled Map Lattices. In: Abraham, N.B., Albano, A.M., Passamante, A., Rapp, P.E. (eds) Measures of Complexity and Chaos. NATO ASI Series, vol 208. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0623-9_57
Download citation
DOI: https://doi.org/10.1007/978-1-4757-0623-9_57
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-0625-3
Online ISBN: 978-1-4757-0623-9
eBook Packages: Springer Book Archive