Data Requirements for Reliable Estimation of Correlation Dimensions

  • A. M. Albano
  • A. I. Mees
  • G. C. de Guzman
  • P. E. Rapp
Part of the NATO ASI Series book series (NSSA, volume 138)


It is not always possible to resolve the dimension of an attractor from a finite data set. The number of data points required depends on the structure of the attractor, the distribution of points on the attractor, and the precision of the data. If the chaotic component of a system’s behaviour is sufficiently small relative to its large scale motion, and if orbits seldom visit the region of the attractor with small scale fractal structure, any method will fail to resolve the attractor’s dimension. It is simple to construct abstract mathematical examples that present this behaviour. However, while these limitations should be explicity recognised, it should also be noted that a growing body of empirical experience suggests that experimentally encountered physical and biological systems do not invariably display these behaviours. It is possible to estimate reliably the dimension of these attractors with comparatively small data sets.


Chaotic System Correlation Dimension Wigner Distribution Lorenz Attractor Bryn Mawr 
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  1. [1]
    N.B. Abraham, A.M. Albano, B. Das, G. de Guzman, S. Yong, R.S. Gioggia, G.P. Puccioni and J.R. Tredicce, Phys. Lett. 114A: 217–221 (1985).Google Scholar
  2. [2]
    K. Aihara and G. Matsumoto, in “Chaos”, A.V. Holden, ed., 257–269, University Press, Manchester & Princeton (1986).Google Scholar
  3. [3]
    A.M. Albano, J. Abounadi, T.H. Chyba, C.E. Searle and S. Yong, J. Opt. Soc. Amer. 2B: 47–55 (1985).CrossRefGoogle Scholar
  4. [4]
    A.M. Albano et al., in “Dimension and Entropies in Chaotic Systems. Quantification of Complex Behaviour”, G. Mayer-Kress, ed., 231240, Springer-Verlag, Berlin (1986).Google Scholar
  5. [5]
    D.S. Broomhead and G.P. King, Physica 2OD: 217–236 (1986a).Google Scholar
  6. [6]
    D.S. Broomhead and G.P. King, in “Nonlinear Phenomena and Chaos”, S. Sarkan, ed., Adam Hilger, Bristol (1986b).Google Scholar
  7. [7]
    W.E. Caswell and J.A. Yorke, in “Dimensions and Entropies in Chaotic Systems”, G. Mayer-Kress, ed., 123–136, Springer-Verlag, Berlin (1986).CrossRefGoogle Scholar
  8. [8]
    T.F. Chan, ACM Trans. Math. Software 8: 84–88 (1982).CrossRefGoogle Scholar
  9. [9]
    T.A.C.M. Claasen and W.F.G. Mecklenbrauker, Phillips J. Res. 35: 217–250 (1980).Google Scholar
  10. [10]
    T.A.C.M. Claasen and W.F.G. Mecklenbrauker, Phillips J. Res. 35: 276–300 (1980).Google Scholar
  11. [11]
    T.A.C.M. Claasen and W.F.G. Mecklenbrauker, Phillips J. Res. 35: 372–389 (1980).Google Scholar
  12. [12]
    J.-P. Eckmann and D. Ruelle, Rev. modn. Phys. 54: 617–656 (1985).CrossRefGoogle Scholar
  13. [13]
    J.D. Farmer, in “Evolution of Order and Chaos”, H. Haken, ed., 228–246, Springer-Verlag, Berlin (1982a).Google Scholar
  14. [14]
    J.D. Farmer, Z. Naturforsch. 37A: 1304–1325 (1982b).Google Scholar
  15. [15]
    J.D. Farmer, E. Ott and J.A. Yorke, Physica 7D: 153–180 (1983).Google Scholar
  16. [16]
    A.M. Fraser and H.L. Swinney, Phys. Rev. 33A: 1134–1140 (1986).Google Scholar
  17. [17]
    L. Glass, A. Shrier and J. Belair, in “Chaos”, A.V. Holden, ed., 237–256, University Press, Manchester & Princeton (1986).Google Scholar
  18. [18]
    G. Golub and W. Kahan, SIAM J. Numer. Anal. 2: 205–224 ( (1965).Google Scholar
  19. [19]
    G.B. Golub and C. Reinsch, Numer. Math. 14: 403–420 (1970).CrossRefGoogle Scholar
  20. [20]
    G.H. Golub and C.F. Van Loan, “Matrix Computations”, Johns Hopkins University Press, Baltimore (1983).Google Scholar
  21. [21]
    P. Grassberger and I. Procaccia, Physica 9D: 189–208 (1983a).Google Scholar
  22. [22]
    P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50: 346–349 (1983b).CrossRefGoogle Scholar
  23. [23]
    P. Grassberger and I. Procaccia, Phys. Rev. A 28A: 2591–2593 (1983c).CrossRefGoogle Scholar
  24. [24]
    M.R. Guevara, L. Glass and A. Shrier, Science, Wash. 214: 1350–1353 (1981).CrossRefGoogle Scholar
  25. [25]
    H. Hayashi, S. Ishizuka and K. Hirakawa, Phys. Lett. A 98A: 474–476 (1983).CrossRefGoogle Scholar
  26. [26]
    H. Hayashi, S. Ishizuka, M. Ohta and K. Hirakawa, Phys. Lett. 88A: 435–438 (1983).Google Scholar
  27. [27]
    M. Hénon, Commun. math. Phys. 50: 69–78 (1976).CrossRefGoogle Scholar
  28. [28]
    H.G.E. Hentschel and I. Procaccia, Physica 8D: 435–444 (1983).Google Scholar
  29. [29]
    M.H. Jensen, L.P. Kadanoff, A. Libchaber, I. Procaccia and J. Stavans, Phys. Rev. Lett. 55: 2798–2801 (1986).CrossRefGoogle Scholar
  30. [30]
    J. Kurth and H. Herzel, Physica D, submitted (1986).Google Scholar
  31. [31]
    M. Markus and B. Hess, Proc. natn. Acad. Sci. U.S.A. 81: 4394–4398 (1984).CrossRefGoogle Scholar
  32. [32]
    M. Markus, D. Kuschmitz and B. Hess, FEBS Lett. 172: 235–238 (1984).PubMedCrossRefGoogle Scholar
  33. [33]
    J.L. Martiel and A. Goldbeter, Nature, Lond. 313: 590–592 (1985).CrossRefGoogle Scholar
  34. [34]
    A.I. Mees, P.E. Rapp and L.S. Jennings, “Singular value decomposition and embedding dimension”, Phys. Rev., in press.Google Scholar
  35. [35]
    N.H. Morgan and A.S. Gevins, IEEE Trans. Biomed. Eng. BME-33: 6670 (1986).Google Scholar
  36. [36]
    C. Nicolis and G. Nicolis, Nature, Lond. 311: 529–532 (1984).CrossRefGoogle Scholar
  37. [37]
    P.E. Rapp, I.D. Zimmerman, A.M. Albano, G.C. de Guzman and N.N. Greenbaum, Phys. Lett. 110A: 335–338 (1985a).CrossRefGoogle Scholar
  38. [38]
    P.E. Rapp, I.D. Zimmerman, A.M. Albano, G.C. de Guzman, N.N. Greenbaum and T.R. Bashore, in “Nonlinear Oscillations in Chemistry and Biology”, H.G. Othmer, ed., Springer-Verlag, NY (1985b).Google Scholar
  39. [39]
    W.M. Schaffer, IMA J. Maths. appl. Med. and Biol. 2: 221–252 (1985).CrossRefGoogle Scholar
  40. [40]
    C.W. Simm, M.L. Sawley, F. Skiff and A. Pochelon, “On the analysis of experimental signals for evidence of determinstic chaos”, preprint (1986).Google Scholar
  41. [41]
    F. Takens, in “Dynamical Systems aid Turbulence, Lecture Notes in Mathematics, Volume 898”, D.A. Rand and L.S. Young, eds., 365381, Springer-Verlag, NY (1980).Google Scholar
  42. [42]
    J. Theiler, Phys. Rev. A 34: 2427 (1986).PubMedCrossRefGoogle Scholar
  43. [43]
    J. Vastano and E.J. Kostelich, in “Dimensions and Entropies in Chaotic Systems”, G. Mayer-Kress, ed., 100–107, Springer-Verlag, Berlin (1986).CrossRefGoogle Scholar
  44. [44]
    S. Watanabe, Proc. Conference on Information Theory, Prague (1965).Google Scholar
  45. [45]
    H. Whitney, Ann. Math. 37: 645–680 (1936).CrossRefGoogle Scholar
  46. [46]
    E. Wigner, Phys. Rev. 40: 749–759 (1932).CrossRefGoogle Scholar
  47. [47]
    A. Wolf, J.B. Swift, H.L. Swinney and J.A. Vastano, Physica 16D: 285–317 (1985).Google Scholar

Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • A. M. Albano
    • 1
  • A. I. Mees
    • 2
  • G. C. de Guzman
    • 3
  • P. E. Rapp
    • 4
  1. 1.Department of PhysicsBryn Mawr CollegeBryn MawrUSA
  2. 2.Department of MathematicsUniversity of Western AustraliaNedlandsUSA
  3. 3.Center for Complex SystemsFlorida Atlantic UniversityBoca RatonUSA
  4. 4.Department of Physiology and BiochemistryMedical College of PennsylvaniaPhiladelphiaUSA

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