Journal of Statistical Physics

, Volume 65, Issue 3–4, pp 579–616 | Cite as


  • Tim Sauer
  • James A. Yorke
  • Martin Casdagli


Mathematical formulations of the embedding methods commonly used for the reconstruction of attractors from data series are discussed. Embedding theorems, based on previous work by H. Whitney and F. Takens, are established for compact subsetsA of Euclidean space Rk. Ifn is an integer larger than twice the box-counting dimension ofA, then almost every map fromR k toR n , in the sense of prevalence, is one-to-one onA, and moreover is an embedding on smooth manifolds contained withinA. IfA is a chaotic attractor of a typical dynamical system, then the same is true for almost everydelay-coordinate map fromR k toR n . These results are extended in two other directions. Similar results are proved in the more general case of reconstructions which use moving averages of delay coordinates. Second, information is given on the self-intersection set that exists whenn is less than or equal to twice the box-counting dimension ofA.

Key words

Embedding chaotic attractor attractor reconstruction probability one prevalence box-counting dimension delay coordinates 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Abarbanel, R. Brown, and J. Kadtke, Prediction in chaotic nonlinear systems: Methods for time series with broadband Fourier spectra, preprint.Google Scholar
  2. 2.
    A. M. Albano, J. Muench, C. Schwartz, A. Mees, and P. Rapp, Singular value decomposition and the Grassberger-Procaccia algorithm,Phys. Rev. A 38:3017–3026 (1988).Google Scholar
  3. 3.
    V. I. Arnold,Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, New York, 1983).Google Scholar
  4. 4.
    R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, and M. A. Rubio, Dimension increase in filtered chaotic signals,Phys. Rev. Lett. 60:979–982 (1988).Google Scholar
  5. 5.
    D. S. Broomhead and G. P. King, Extracting qualitative dynamics from experimental data,Physics 20D:217–236 (1986).Google Scholar
  6. 6.
    M. Casdagli, Nonlinear prediction of chaotic time series,Physica 35D:335–356 (1989).Google Scholar
  7. 7.
    M. Casdagli, S. Eubank, D. Farmer, and J. Gibson, State-space reconstruction in the presence of noise, preprint.Google Scholar
  8. 8.
    W. Ditto, S. Rauseo, and M. Spano, Experimental control of chaos,Phys. Rev. Lett. 65:3211–3214 (1990).Google Scholar
  9. 9.
    J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,Rev. Mod. Phys. 57:617–656 (1985).Google Scholar
  10. 10.
    A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Hölder continuity for the inverse of Mañe's projection,Comptes Rendus, to appear.Google Scholar
  11. 11.
    K. Falconer,Fractal Geometry (Wiley, New York, 1990).Google Scholar
  12. 12.
    J. D. Farmer and J. Sidorowich, Predicting chaotic time series,Phys. Rev. Lett. 59:845–848 (1987).Google Scholar
  13. 13.
    J. D. Farmer and J. Sidorowich, Exploiting chaos to predict the future and reduce noise, Technical Report LA-UR-88-901, Los Alamos National Laboratory (1988).Google Scholar
  14. 14.
    G. Golub and C. Van Loan,Matrix Computations, 2nd ed. (Johns Hopkins University Press, Baltimore, Maryland, 1989).Google Scholar
  15. 15.
    E. Kostelich and J. Yorke, Noise reduction: Finding the simplest dynamical system consistent with the data,Physica 41D:183–196 (1990).Google Scholar
  16. 16.
    E. Kostelich and J. Yorke, Noise reduction in dynamical systems,Phys. Rev. A 38:1649–1652 (1988).Google Scholar
  17. 17.
    R. Mañé, On the dimension of the compact invariant sets of certain nonlinear maps, inLecture Notes in Mathematics, No. 898 (Springer-Verlag, 1981).Google Scholar
  18. 18.
    P. Marteau and H. Abarbanel, Noise reduction in chaotic time series using scaled probabilistic methods, preprint.Google Scholar
  19. 19.
    P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes,Ann. Acad. Sci. Fenn. Math. 1:227–224 (1975).Google Scholar
  20. 20.
    F. Mitschke, M. Möller, and W. Lange, Measuring filtered chaotic signals,Phys. Rev. A 37:4518–4521 (1988).Google Scholar
  21. 21.
    N. Packard, J. Crutchfield, D. Farmer, and R. Shaw, Geometry from a time series,Phys. Rev. Lett. 45:712 (1980).Google Scholar
  22. 22.
    W. Rudin,Real and Complex Analysis, 2nd ed. (McGraw-Hill, New York, 1974).Google Scholar
  23. 23.
    J.-C. Roux and H. Swinney, Topology of chaos in a chemical reaction, inNonlinear Phenomena in Chemical Dynamics, C. Vidal and A. Pacault, eds. (Springer, Berlin, 1981).Google Scholar
  24. 24.
    B. Hunt, T. Sauer and J. Yorke, Prevalence: A translation-invariant “almost every” on infinite-dimensional spaces, preprint.Google Scholar
  25. 25.
    T. Sauer and J. Yorke, Statistically self-similar sets, preprint.Google Scholar
  26. 26.
    J. Sommerer, W. Ditto, C. Grebogi, E. Ott, and M. Spano, Experimental confirmation of the theory for critical exponents of crises,Phys. Lett. A 153:105–109 (1991).Google Scholar
  27. 27.
    F. Takens, Detecting strange attractors in turbulence, inLecture Notes in Mathematics, No. 898 (Springer-Verlag, 1981).Google Scholar
  28. 28.
    B. Townshend, Nonlinear prediction of speech signals, preprint.Google Scholar
  29. 29.
    H. Whitney, Differentiable manifolds,Ann. Math. 37:645–680 (1936).Google Scholar
  30. 30.
    J. Yorke, Periods of periodic solutions and the Lipschitz constant,Proc. Am. Math. Soc. 22:509–512 (1969).Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Tim Sauer
    • 1
  • James A. Yorke
    • 2
  • Martin Casdagli
    • 3
  1. 1.Department of Mathematical SciencesGeorge Mason UniversityFairfax
  2. 2.Institute of Physical Science and TechnologyUniversity of MarylandCollege Park
  3. 3.Santa Fe InstituteSanta Fe

Personalised recommendations