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Determination of Attractor Dimension and Entropy for Various Flows: An Experimentalist’s Viewpoint

  • J. G. Caputo
  • B. Malraison
  • P. Atten
Part of the Springer Series in Synergetics book series (SSSYN, volume 32)

Abstract

In order to characterise quantitatively the behaviour of dissipative dynamical systems we have to determine the values of information dimension, metric entropy and Lyapunov exponents associated with the limit sets in phase space. For numerically integràble dynamical systems such as iterated maps and systems of ordinary differential equations, methods are available which lead to the determination of Lyapunov exponents with an accuracy generally depending only on the power of the utilized computer. We furthermore have the values of information dimension and metric entropy by applying the conjectured formulas relating their values to the Lyapunov exponents.

Keywords

Lyapunov Exponent Lorenz System Information Dimension Local Slope Convection Regime 
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.

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References

  1. [1]
    J.A. Vastano, Private Communication, and this Conference.Google Scholar
  2. [2]
    M. Sano and Y. Sawada, Phys. Rev. Lett., 55, p. 1082 (1985).MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    H. Froehling, J.P. Crutchfield, D. Farmer, N.H. Packard and R. Shaw, Physica, 3 D, p. 605 (1981).Google Scholar
  4. [4]
    F. Takens in “Dynamical Systems and Turbulence”, Lecture Notes in Math., 898, Springer, Berlin (1981).Google Scholar
  5. [5]
    P. Grassberger and I. Procaccia, Phys. Rev. Lett., 50, p. 346 (1983).MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    P. Grassberger and I. Procaccia, Phys. Rev. A, 28, N° 4, p. 2591, October 1983.ADSCrossRefGoogle Scholar
  7. [7]
    D. Eckman, J.P. Ruelle, Rev. Mod. Phys., 57, July 1985.Google Scholar
  8. [8]
    P. Atten, J.G. Caputo, B. Malraison and Y. Gagne, Journal de Mécanique Théorique et Appliquée, Special Issue (1984), p. 133–156.MathSciNetGoogle Scholar
  9. [9]
    J.G. Caputo and P. Atten, to be published.Google Scholar
  10. [10]
    F. Takens “On the numerical determination of the dimension of an attractor”, pre-print.Google Scholar
  11. [11]
    P. Grassberger and I. Procaccia, Physica, 13 D, p. 34–54 (1984).MathSciNetMATHGoogle Scholar
  12. [12]
    M. Dubois, P. Bergé and V. Croquette, J. Physique Lett., 43, p. L-295–L-98 (1982).CrossRefGoogle Scholar
  13. [13]
    B. Malraison, P. Atten, P. Bergé and M. Dubois, J. Physique Lett., 44, p. L-897–L-902 (1983).CrossRefGoogle Scholar
  14. [14]
    V. Franceschini, Physica., 6 D, p. 285 (1983).MathSciNetMATHGoogle Scholar
  15. [15]
    R.K. Tavakol, and A.S. Tworkowski, Physics Letters, 102 A, p. 273 (1984).MathSciNetCrossRefGoogle Scholar
  16. [16]
    B. Malraison and P. Atten in “Symmetries and broken symmetries”, Pub. N. Boccara (IDSET, Paris), p. 439 (1981).Google Scholar
  17. [17]
    B. Malraison and P. Atten, Phys. Rev. Lett., 49, p. 273 (1982).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • J. G. Caputo
    • 1
  • B. Malraison
    • 1
  • P. Atten
    • 1
  1. 1.Laboratoire d’Electrostatique et de Materiaux DielectriquesCentre National de la Recherche ScientifiqueGrenoble CédexFrance

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