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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.

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  1. H. Abarbanel, R. Brown, and J. Kadtke, Prediction in chaotic nonlinear systems: Methods for time series with broadband Fourier spectra, preprint.

  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. V. I. Arnold,Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, New York, 1983).

    Google Scholar 

  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. D. S. Broomhead and G. P. King, Extracting qualitative dynamics from experimental data,Physics 20D:217–236 (1986).

    Google Scholar 

  6. M. Casdagli, Nonlinear prediction of chaotic time series,Physica 35D:335–356 (1989).

    Google Scholar 

  7. M. Casdagli, S. Eubank, D. Farmer, and J. Gibson, State-space reconstruction in the presence of noise, preprint.

  8. W. Ditto, S. Rauseo, and M. Spano, Experimental control of chaos,Phys. Rev. Lett. 65:3211–3214 (1990).

    Google Scholar 

  9. J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,Rev. Mod. Phys. 57:617–656 (1985).

    Google Scholar 

  10. A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Hölder continuity for the inverse of Mañe's projection,Comptes Rendus, to appear.

  11. K. Falconer,Fractal Geometry (Wiley, New York, 1990).

    Google Scholar 

  12. J. D. Farmer and J. Sidorowich, Predicting chaotic time series,Phys. Rev. Lett. 59:845–848 (1987).

    Google Scholar 

  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).

  14. G. Golub and C. Van Loan,Matrix Computations, 2nd ed. (Johns Hopkins University Press, Baltimore, Maryland, 1989).

    Google Scholar 

  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. E. Kostelich and J. Yorke, Noise reduction in dynamical systems,Phys. Rev. A 38:1649–1652 (1988).

    Google Scholar 

  17. R. Mañé, On the dimension of the compact invariant sets of certain nonlinear maps, inLecture Notes in Mathematics, No. 898 (Springer-Verlag, 1981).

  18. P. Marteau and H. Abarbanel, Noise reduction in chaotic time series using scaled probabilistic methods, preprint.

  19. P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes,Ann. Acad. Sci. Fenn. Math. 1:227–224 (1975).

    Google Scholar 

  20. F. Mitschke, M. Möller, and W. Lange, Measuring filtered chaotic signals,Phys. Rev. A 37:4518–4521 (1988).

    Google Scholar 

  21. N. Packard, J. Crutchfield, D. Farmer, and R. Shaw, Geometry from a time series,Phys. Rev. Lett. 45:712 (1980).

    Google Scholar 

  22. W. Rudin,Real and Complex Analysis, 2nd ed. (McGraw-Hill, New York, 1974).

    Google Scholar 

  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. B. Hunt, T. Sauer and J. Yorke, Prevalence: A translation-invariant “almost every” on infinite-dimensional spaces, preprint.

  25. T. Sauer and J. Yorke, Statistically self-similar sets, preprint.

  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. F. Takens, Detecting strange attractors in turbulence, inLecture Notes in Mathematics, No. 898 (Springer-Verlag, 1981).

  28. B. Townshend, Nonlinear prediction of speech signals, preprint.

  29. H. Whitney, Differentiable manifolds,Ann. Math. 37:645–680 (1936).

    Google Scholar 

  30. J. Yorke, Periods of periodic solutions and the Lipschitz constant,Proc. Am. Math. Soc. 22:509–512 (1969).

    Google Scholar 

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Sauer, T., Yorke, J.A. & Casdagli, M. Embedology. J Stat Phys 65, 579–616 (1991).

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Key words

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