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Analyzing Periodic Saddles in Experimental Strange Attractors

  • Daniel P. Lathrop
  • Eric J. Kostelich
Part of the NATO ASI Series book series (NSSB, volume 208)

Abstract

This paper discusses a way to locate and analyze the periodic orbits in an attractor reconstructed from time series data, and the technique is applied to data from an experiment on the Belousov-Zhabotinskii chemical reaction. The topological entropy of the attractor is estimated by approximating the dynamics with a subshift of finite type. The Lyapunov exponents are computed from the data using a method suggested by Eckmann and Ruelle and agree well with the estimated topological entropy and information dimension.

Keywords

Periodic Orbit Lyapunov Exponent Finite Type Topological Entropy Recurrence Time 
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]
    F. Takens, in Dynamical Systems and Turbulence, ed. by D. A. Rand and L.-S. Young, Springer Lecture Notes in Mathematics, Vol. 898 ( New York: Springer-Verlag, 1981 ), p. 366.Google Scholar
  2. [2]
    P. Grassberger and I. Procaccia, Physica D 9 (1983), 189.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    R. Badii and A. Politi, J. Stat. Phys. 40 (1985), 725.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57 (1985), 617.MathSciNetCrossRefGoogle Scholar
  5. [5]
    A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Physica D 16 (1985), 285.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    E. J. Kostelich and J. A. Yorke, Phys. Rev. A 38 (1988), 1649.MathSciNetCrossRefGoogle Scholar
  7. [7]
    E. J. Kostelich and J. A. Yorke, submitted to Physica D.Google Scholar
  8. [8]
    D. P. Lathrop and E. J. Kostelich, Phys. Rev. A, in press.Google Scholar
  9. [9]
    J. C. Roux, R. H. Simoyi, and H. L. Swinney, Physica D 8 (1983), 257.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    P. Richetti, P. De Kepper, J. C. Roux, and H. L. Swinney, J. Stat. Phys. 48 (1987), 977.MATHCrossRefGoogle Scholar
  11. [11]
    F. Argoul, A. Arneodo, P. Richetti, J. C. Roux, and H. L. Swinney, Acc. Chem. Res. 86 (1987), 119.Google Scholar
  12. [12]
    D. Auerbach, P. Cvitanovié, J.-P. Eckmann, G. H. Gunaratne, and I. Procaccia, Phys. Rev. Lett. 58 (1987), 2387.MathSciNetCrossRefGoogle Scholar
  13. [13]
    P. Cvitanovié, Phys. Rev. Lett. 61 (1988), 2729.Google Scholar
  14. [14]
    C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. A 36 (1988), 3522.MathSciNetCrossRefGoogle Scholar
  15. [15]
    A. M. Fraser and H. L. Swinney Phys. Rev. A 33 (1986), 1134.Google Scholar
  16. [16]
    C. Grebogi, E. Ott and J. A. Yorke, Physica D 7 (1983), 187.MathSciNetCrossRefGoogle Scholar
  17. [17]
    P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems ( Boston: Birkhäuser, 1981 ).Google Scholar
  18. [18]
    P. Cvitanovié, G. Gunaratne, and I. Procaccia, Phys. Rev. A 38 (1988), 1503.MathSciNetCrossRefGoogle Scholar
  19. [19]
    R. L. Devaney, An Introduction to Chaotic Dynamical Systems (Menlo Park, CA: The Benjamin/Cummings Publishing Co., Inc., 1986 ).MATHGoogle Scholar
  20. [20]
    J.-P. Eckmann, S. O. Kamphorst, D. Ruelle and S. Ciliberto, Phys. Rev. A 34 (1986), 4971.MathSciNetCrossRefGoogle Scholar
  21. [21]
    M. Sano and Y. Sawada, Phys. Rev. Lett. 55 (1985), 1082.MathSciNetCrossRefGoogle Scholar
  22. [22]
    J. L. Kaplan and J. A. Yorke, “Chaotic Behavior of Multidimensional Difference Equations,” in Functional Differential Equations and Approximations of Fixed Points, ed. by H. O. Peitgen and H. O. Walther, Springer Lecture Notes in Mathematics Vol. 730 ( New York: Springer-Verlag, 1979 ).Google Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • Daniel P. Lathrop
    • 1
  • Eric J. Kostelich
    • 1
    • 2
  1. 1.Center for Nonlinear Dynamics and Department of PhysicsUniversity of TexasAustinUSA
  2. 2.Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA

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