Systematic Errors in Estimating Dimensions from Experimental Data

  • W. Lange
  • M. Möller
Part of the NATO ASI Series book series (NSSB, volume 208)


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 [1] that requires only a single-variable time series by making use of the embedding technique originally proposed by Takens [2] 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.


Lyapunov Exponent Strange Attractor Chaotic Signal Lyapunov Dimension Average Length Scale 
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  1. 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. 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. 3.
    J.G. Caputo, “Determination of Attractor Dimension and Entropy for Various Flows: An Experimentalists Viewpoint”, in [SI, p.180Google Scholar
  4. 4.
    R. Badii and A. Politi, Phys. Rev. A 35, 1288 (1987); J.G. Caputo and P. Atten, “Metric entropy: An experimental means for characterizing and quantifying chaos”, Phys. Rev. A 35, 1311 (1987);MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dimensions and Entropies in Dynamical Systems,ed. by G. Mayer-Kress, (Springer, Berlin 1986)Google Scholar
  6. 6.
    L.A. Smith, “Intrinsic Linits on Dimension Calculations” Phys. Lett. A133, 283 (1988)CrossRefGoogle Scholar
  7. 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
  8. 8.
    R.L. Somorjai, M.K. Ali, “An efficient algorithm for estimating dimensionalities”, Can. J. Chem. 66, 979 (1988)CrossRefGoogle Scholar
  9. 9.
    A. Ben Mizrachi, I. Procaccia, and P. Grassberger, “Characterization of experimental (noisy) strange attractors”, Phys. Rev. A 29, 975 (1984).CrossRefGoogle Scholar
  10. 10.
    R. Badii and A. Politi, “On the Fractal Dimension of Filtered Chaotic Signals”, in [5], p.67Google Scholar
  11. 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
  12. 12.
    F. Mitschke, M. Möller, W. Lange, “Measuring Filtered Chaotic Signals”, Phys. Rev. A, 37, 4518 (1988)CrossRefGoogle Scholar
  13. 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
  14. 14.
    R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi and M.A. Rubio, “Dimension Increase in Filtered Chaotic Signals”, Phys. Rev. Lett 60, 979 (1988).CrossRefGoogle Scholar
  15. 15.
    F. Mitschke, N. Flüggen, “Chaotic Behaviour of a Hybrid Optical Bistable System without a Time Delay”, Appl. Phys. B35, 59 (1984)CrossRefGoogle Scholar
  16. 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. 17.
    F. Mitschke (private communication)Google Scholar
  18. 18.
    N.S. Jayant and P. Noll, Digital Coding of Waveforms, ( Prentice-Hall, Englewood Cliffs, NJ, 1984 ).Google Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • W. Lange
    • 1
  • M. Möller
    • 1
  1. 1.Institut für Angewandte PhysikWestfälischen Wilhelms-Universität MünsterMünsterFederal Republic of Germany

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