Symbolic Dynamics from Chaotic Time Series

  • A. Destexhe
  • G. Nicolis
  • C. Nicolis
Part of the NATO ASI Series book series (NSSB, volume 208)


Following the ideas of Ruelle and others [1,2], an embedding phase space can be reconstructed from experimental systems on the basis of time series data. The introduction of numerical methods for calculating dimensions, entropies, Lyapunov exponents and other related properties, has permitted extensive investigations of chaotic experimental systems these last years. However, severe restrictions about the applicability of these methods were noticed, especially for high dimensional systems [3–6].


Markov Process Lyapunov Exponent Chaotic Dynamic Symbolic Dynamic Chaotic Time Series 
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  1. [1]
    Eckmann J.P. and Ruelle D., Rev. Mod. Phys. 57 (1985), 617MathSciNetCrossRefGoogle Scholar
  2. [2]
    Packard N.H., Crutchfield J.P., Farmer J.D. and Shaw R.S.,Phys. Rev. Lett. 45 (1980), 712 F.Takens: in Dynamical systems and turbulence. Eds D.A.Rand & L.S.Young. Lect. Notes in Math. 898, 366 ( Springer, Berlin, 1981 )Google Scholar
  3. [3]
    Theiler J., Phys. Rev. A 34 (1986) 2427, and references therein.Google Scholar
  4. [4]
    Babloyantz A. and Destexhe A., in Temporal disorder in human oscillatory systems Eds. L. Rensing, U. an der Heiden and M.C. Mackey, Springer Series in Synergetics 36,48 ( Springer, Berlin, 1987 )Google Scholar
  5. [5]
    Destexhe A., Sepulchre J.A. and Babloyantz A., Phys. Lett. A 132 (1988), 101.CrossRefGoogle Scholar
  6. [6]
    Eckmann J.P. and Ruelle D., preprint, (1989)Google Scholar
  7. [7]
    Aizawa Y., Prog. Theor. Phys. 70 (1983), 1249zbMATHGoogle Scholar
  8. [8]
    Feller W., in: An Introduction to Probability Theory and its Applications ( Wiley, New York, 1957 )Google Scholar
  9. [9]
    Nicolis G.and Nicolis C., Phys. Rev. A. 38 (1988), 427CrossRefGoogle Scholar
  10. [10]
    Rössler O.E., Ann. N.Y. Acad. Sci. 316 (1979) 376CrossRefGoogle Scholar
  11. [11]
    Nicolis G., Rao G.S., Rao J.S. and Nicolis C., in Structure, Coherence and Chaos in Dynamical Systems, Eds. Christiaensen P.L. and Parmentier R.D. ( Manchester University Press, 1989 )Google Scholar
  12. [12]
    Kemeny J.G., Snell J.L. and Knapp A.W., in Denumerable Markov Chains, Graduate texts in Mathematics ( Springer, Berlin, 1976 )Google Scholar
  13. [13]
    A. Destexhe, Phys. Lett. A, in press (1989)Google Scholar
  14. [14]
    Babloyantz A. and Destexhe A., Biological Cybernetics 58 (1988), 203MathSciNetCrossRefGoogle Scholar
  15. [15]
    an almost exhaustive state of the art can be found in: Dynamics of Sensory and Cognitive Processing by the Brain Ed. E. Basar, Springer Series in Brain Dynamics vol.1 (1988), vol.2 (1989).Google Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • A. Destexhe
    • 1
  • G. Nicolis
    • 1
  • C. Nicolis
    • 2
  1. 1.Service de Chimie PhysiqueUniversité Libre de BruxellesBruxellesBelgium
  2. 2.Institut d’Aeronomie Spatiale de Belgique1180 BruxellesBelgium

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