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Intermediate Length Scale Effects in Lyapunov Exponent Estimation

  • A. Wolf
  • J. A. Vastano
Part of the Springer Series in Synergetics book series (SSSYN, volume 32)

Abstract

Algorithms for estimating Lyapunov exponents from experimental data monitor the divergence of nearby phase space orbits. These algorithms rely on the assumption that the dynamics on intermediate length scales are “close” to the dynamics on infinitesimal length scales. We have studied two one-dimensional maps in which intermediate length scale dynamics may result in inaccurate exponent estimates. This effect is found to be small enough so that the exponent estimates are still good characterizations of the systems. Similar effects are likely to be present whenever a finite quantity of data is used for Lyapunov exponent estimation.

Keywords

Fractal Dimension Lyapunov Exponent Chaotic System Large Lyapunov Exponent Initial Separation 
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.
    A. Wolf, J. Swift, H. L. Swinney, and J. A. Vastano, Physica 16D, 285 (1985).MathSciNetADSGoogle Scholar
  2. 2.
    J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    More precisely we should say that the long-term growth rate of δx(t) is λ1 For the 1−D map these definitions are equivalent, but in higher dimensions the transient behavior of the vector δx(t) will almost always depend on other Lyapunov exponents.Google Scholar
  4. 4.
    A. Wolf, in Nonlinear Science: Theory and Applications, ed. by A. Holden, (Manchester University Press, 1986).Google Scholar
  5. 5.
    A. Brandstater, J. Swift, H. L. Swinney, A. Wolf, J. D. Farmer, E. Jen, and J. P. Crutch field, Phys. Rev. Lett. 51, 1442 (1983).MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    By intermediate length scales we mean length scales above the noise level but small compared to the overall size of the attractor. We recommend either imposing a small-distance spatial cutoff throughout exponent calculations, low-pass filtering the data, or repeating calculations with the least significant data bit(s) truncated, to test the stability of exponent values.Google Scholar
  7. 7.
    M. Sano and Y. Sawada, Phys. Rev. Lett. 55, 1082 (1985).MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    J. A. Vastano and E. J. Kostelich, in Dimensions and Entropies in Chaotic Systems - Quantification of Complex Behavior, ed. by G. Meyer-Kress (Springer, 1986).Google Scholar
  9. 9.
    E. N. Lorenz, J. Atmos. Sci. 20, 130 (1973).ADSCrossRefGoogle Scholar
  10. 10.
    O. E. Rossler, Phys. Lett. 57A, 397 (1976).ADSGoogle Scholar
  11. 11.
    It would appear that after n iterations there could be 2 n possible ensembles. However, the algebra is such that there are only two possible ensembles after any number of iterations. This is shown in Fig. 2.Google Scholar
  12. 12.
    H. S. Greenside, A. Wolf, J. Swift, and T. Pignataro, Phys. Rev. A25, 3453 (1982).MathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • A. Wolf
    • 1
  • J. A. Vastano
    • 2
  1. 1.The Cooper UnionSchool of EngineeringNew YorkUSA
  2. 2.Department of Physics and Center for Nonlinear DynamicsUniversity of TexasAustinUSA

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