Systematic Errors in Estimating Dimensions from Experimental Data
The necessity of characterizing chaotic dynamics quantitatively by assigning to them measures like the dimension of the underlying attractor or the entropy production of the system has generally been accepted. The corresponding methods are practiced now by many experimentalists. Most of them have adopted the method by Grassberger and Procaccia  that requires only a single-variable time series by making use of the embedding technique originally proposed by Takens  Though the validity of the method is beyond any doubt, its practical application presents problems, since it relies on assumptions which are not generally fulfilled in the experiment.
KeywordsLyapunov Exponent Strange Attractor Chaotic Signal Lyapunov Dimension Average Length Scale
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- 1.P. Grassberger and I. Procaccia,“Characteriziation of Strange Attractors”, Phys. Rev. Lett. 50, 346 (1983); “Measuring the Strangeness of Strange Attractors”, Physica 9D, 189 (1983); “Estimation of the Kolmogorov entropy from a chaotic signal”, Phys. Rev. A 28, 2591 (1983); “Dimensions and Entropies of Strange Attractors from a Fluctuating Dynamics Approach”, Physica 13D, 34 (1984)MathSciNetzbMATHGoogle Scholar
- 2.F. Takens, “Detecting Strange Attractors in Turbulence”, in Dynamical Systems and Turbulence, Vol.898 of Lecture Notes in Mathematics, ed. by D.A. Rand and L.S. Young ( Springer, Berlin; 1981 )Google Scholar
- 3.J.G. Caputo, “Determination of Attractor Dimension and Entropy for Various Flows: An Experimentalists Viewpoint”, in [SI, p.180Google Scholar
- 5.Dimensions and Entropies in Dynamical Systems,ed. by G. Mayer-Kress, (Springer, Berlin 1986)Google Scholar
- 7.A.M. Albano, J. Muench, C. Schwartz, A.I. Mees and P.E. Rapp, “Singular-value decomposition and the Grassberger-Procaccia algorithm”, Phys. Rev. A 38, 3017 (1988), R. Badii and A. Politi, J. Stat. Phys. 40, 725 (1985), G. Broggi, J. Opt. Soc. Am. BS, 1020 (1988)Google Scholar
- 10.R. Badii and A. Politi, “On the Fractal Dimension of Filtered Chaotic Signals”, in , p.67Google Scholar
- 11.F. Mitschke, M. Möller, W. Lange, “On Systematic Errors in Characterizing Chaos”, in Optical Bistability IV,ed. by W. Firth, N. Peyghambarian and A. Tallet, (les Editions Physique, Paris, 1988), p. C2–397 [reprinted from J. Phys. Colloq. 49, suppl.6, C2–397 (1988)].Google Scholar
- 13.M. Möller, W. Lange, F. Mitschke, N.B. Abraham, U. Hübner, “Errors from Digitizing and Noise in Estimating Attractor Dimensions”, Phys. Lett. A (to appear)Google Scholar
- 16.O. Chennaoui, W. Liebert, K. Pawelzik, H.G. Schuster, “Filterinversion bei chaotischen Zeitreihen”, Verhandl. DPG (VI) 24, 4(1989), DY 12–54Google Scholar
- 17.F. Mitschke (private communication)Google Scholar
- 18.N.S. Jayant and P. Noll, Digital Coding of Waveforms, ( Prentice-Hall, Englewood Cliffs, NJ, 1984 ).Google Scholar