Time Evolution of Local Complexity Measures and Aperiodic Perturbations of Nonlinear Dynamical Systems

  • Gottfried Mayer-Kress
  • Alfred Hübler
Part of the NATO ASI Series book series (NSSB, volume 208)

Abstract

We discuss numerical algorithms for estimating dimensional complexity of observed time-series with special emphasis on biological and medical applications. Factors which enter the procedure are discussed and applied to local estimates of pointwise dimensions or crowding indices. We illustrate the concepts with the help of experimental time-series obtained from speech signals. The temporal evolution of the crowding index shows oscillations which can be correlated with properties of the time-series. We compare the time evolution of the dimensional complexity parameter with the original time-series and also with recurrence plots of the embedded time series.

Besides the analysis of spontaneous activity of biological systems it is often more useful to study event related potentials. We have generalized our analysis code in a way that attractors can also be reconstructed from such non contiguous signals. Finally we discuss the possibility of nonlinear, aperiodic stimulation of nonlinear and chaotic systems as a method for very selective excitations of specific nonlinear modes. We discuss possible applications of this method to habituation phenomena and diagnostic use in connection with event-related potentials.

Keywords

Chaotic System Speech Signal Reference Vector Recurrence Plot Dimensional Complexity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Cowan, in “Pattern Formation by Dynamic Systems and Pattern Recognition”, H. Haken (Ed.), Springer Series in Synergetics Vol. 5, Springer Verlag, Berlin Heidelberg, New York, 1978Google Scholar
  2. [2]
    U. Dressler, G. Mayer-Kress, W. Lauterborn, “Local Divergence Rates in Nonlinear Dynamical Systems”, in preparationGoogle Scholar
  3. [3]
    J.P. Eckmann, S. Oliffson Kamphorst, D. Ruelle, Europhysics Letters, 4, 973–977, (1987)CrossRefGoogle Scholar
  4. [4]
    Th. Eisenhammer, A. Hübler, G. Mayer-Kress, P. Milonni “Aperiodic resonant excitation of classical and quantum anharmonic oscillators”, in preparationGoogle Scholar
  5. [5]
    J.D. Farmer, E. Ott, J. Yorke, “The Dimension of Chaotic Attractors”, Physica 7D, 153, (1983)MathSciNetGoogle Scholar
  6. [6]
    J.D. Farmer, J. Sidorowich, “Exploiting Chaos to Predict the Future and Reduce Noise”, Reviews of Modern Physics, (1989)Google Scholar
  7. [7]
    E. Flynn, private communicationGoogle Scholar
  8. [8]
    A.M. Fraser, H.L. Swinney, Phys. Rev. A 33, 1134–1140; (1986)MathSciNetMATHGoogle Scholar
  9. [9]
    P.Grassberger, I. Procaccia, “Measuring the Strangeness of Strange Attractors”, Physica 9D, 189, (1983)MathSciNetMATHGoogle Scholar
  10. [10]
    H. Haken, “Advanced Synergetics”, Springer, Berlin 1983MATHGoogle Scholar
  11. [11]
    H. Haken, A. Fuchs, in: “Neural and Synergetic Computers”, H. Haken (Ed.), Springer Series in Synergetics Vol. 42, Springer Verlag, Berlin, Heidelberg, New-York, 1988Google Scholar
  12. [12]
    J. Holzfuss, G. Mayer-Kress, “An Approach to Error-Estimation in the Application of Dimension Algorithms”, in: “Dimensions and Entropies in Chaotic Systems”, G. Mayer-Kress (ed.), Springer Series in Synergetics Vol. 32, Springer Verlag, Berlin etc., 1986Google Scholar
  13. [13]
    A. Hübler, E. Löscher, “Resonant Stimulation and Control of Nonlinear Oscillators”, Naturwissenschaften 76, 67(1989)CrossRefGoogle Scholar
  14. A. Hübler, E. Löscher, “Resonant Stimulation of Complex Systems”, to appear in Helv.Phys.Acta 61 Google Scholar
  15. [14]
    A. Lapedes, private communicationGoogle Scholar
  16. [15]
    Y.C. Lee, G. Mayer-Kress, G. Papcun, unpublished resultsGoogle Scholar
  17. [16]
    G. Mayer-Kress, (ed.), “Dimensions and Entropies in Chaotic Systems”, Springer Series in Synergetics, Vol. 32, Springer-Verlag Berlin, Heidelberg 1986MATHGoogle Scholar
  18. [17]
    G. Mayer-Kress, S.P. Layne, “Dimensionality of the Human Electroencephalogram”, Proc. of the New York Academy of Sciences conf. “Perspectives in Biological Dynamics and Theoretical Medicine”, A.S. Mandell, S. Koslow (eds.)Annals of the New York Academy of Sciences, Vol. 504, New York, 1987Google Scholar
  19. [18]
    G. Mayer-Kress, “Application of Dimension Algorithms to Experimental Chaos”, in: “Directions in Chaos”, Hao Bai-lin (Ed.), World Scientific Publishing Company, Singapore, 1987Google Scholar
  20. [19]
    G. Mayer-Kress, F. E. Yates, L. Benton, M. Keidel, W. Tirsch, S.J. Pöppl, K. Geist, “Dimensional Analysis of Nonlinear Oscillations in Brain, Heart and Muscle”, Mathematical Biosciences 90, 155–182, 1988MathSciNetMATHCrossRefGoogle Scholar
  21. [20]
    J. Nicolis, G. Mayer-Kress, G. Haubs, “Non-Uniform Chaotic Dynamics with Implications to Information Processing”, Z.Naturforsch. 38a, 1157–1169 (1983)MathSciNetMATHGoogle Scholar
  22. [21]
    I. Procaccia, “Characterization of Fractal Measures as Interwoven Sets of Singularities”, in G. Mayer-Kress, (ed.), “Dimensions and Entropies in Chaotic Systems”, Springer Series in Synergetics, Vol. 32, Springer-Verlag Berlin, Heidelberg 1986Google Scholar
  23. [22]
    O. Sacks, “The Man who Mistook his Wife for a Hat”, Harper & Row, New York 1987Google Scholar
  24. [23]
    K. Srinivasan, this volumeGoogle Scholar
  25. [24]
    C. Wagner, W. Stelzel, A. Hübler, E. Löscher, “Resonante Steuerung nichtlinearer Schwinger”, Helv.Phys.Acta 61, 228, (1988)Google Scholar
  26. [25]
    J. P. Zbilut, G. Mayer-Kress, K. Geist, “Dimensional Analysis of Heart Rate Variability in Heart Transplant Recipients”, Mathematical Biosciences, 90, 49–70, (1988)MathSciNetCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • Gottfried Mayer-Kress
    • 1
    • 2
  • Alfred Hübler
    • 3
  1. 1.Mathematics DepartmentUniversity of California at Santa CruzUSA
  2. 2.Department of Chemical Engineering, Princeton University Center for Nonlinear StudiesLos Alamos National Laboratory MS-B258Los AlamosUSA
  3. 3.Center for Complex Systems Research, Department of Physics, Beckman InstituteUniversity of Illinois at Urbana-ChampaignUSA

Personalised recommendations