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**Embedology.**
*(English)*
Zbl 0943.37506

Summary: 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 [Ann. Math. (2) 37, 645-680 (1936; Zbl 0015.32001)] and F. Takens [Lect. Notes Math. 898, 366-381 (1981; Zbl 0513.58032)], are established for compact subsets \(A\) of Euclidean space \(R^k\). If \(n\) is an integer larger than twice the box-counting dimension of \(A\), then almost every map from \(R^k\) to \(R^n\), in the sense of prevalence, is one-to-one on \(A\), and moreover is an embedding on smooth manifolds contained within \(A\). If \(A\) is a chaotic attractor of a typical dynamical system, then the same is true for almost every delay-coordinate map from \(R^k\) to \(R^n\). 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 when \(n\) is less than or equal to twice the box-counting dimension of \(A\).

### MSC:

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

28A80 | Fractals |

57R40 | Embeddings in differential topology |

### Keywords:

embedding; chaotic attractor; attractor reconstruction; probability one; prevalence; box-counting dimension; delay coordinates
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\textit{T. Sauer} et al., J. Stat. Phys. 65, No. 3--4, 579--616 (1991; Zbl 0943.37506)

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### References:

[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). |

[3] | V. I. Arnold,Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, New York, 1983). · Zbl 0507.34003 |

[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). |

[5] | D. S. Broomhead and G. P. King, Extracting qualitative dynamics from experimental data,Physics 20D:217-236 (1986). · Zbl 0603.58040 |

[6] | M. Casdagli, Nonlinear prediction of chaotic time series,Physica 35D:335-356 (1989). · Zbl 0671.62099 |

[7] | M. Casdagli, S. Eubank, D. Farmer, and J. Gibson, State-space reconstruction in the presence of noise, preprint. · Zbl 0736.62075 |

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

[9] | J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,Rev. Mod. Phys. 57:617-656 (1985). · Zbl 0989.37516 |

[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). |

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

[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). · Zbl 0733.65016 |

[15] | E. Kostelich and J. Yorke, Noise reduction: Finding the simplest dynamical system consistent with the data,Physica 41D:183-196 (1990). · Zbl 0705.58036 |

[16] | E. Kostelich and J. Yorke, Noise reduction in dynamical systems,Phys. Rev. A 38:1649-1652 (1988). |

[17] | R. Mañé, On the dimension of the compact invariant sets of certain nonlinear maps, inLecture Notes in Mathematics, No. 898 (Springer-Verlag, 1981). · Zbl 0544.58014 |

[18] | P. Marteau and H. Abarbanel, Noise reduction in chaotic time series using scaled probabilistic methods, preprint. · Zbl 0799.60039 |

[19] | P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes,Ann. Acad. Sci. Fenn. Math. 1:227-224 (1975). · Zbl 0348.28019 |

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

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

[22] | W. Rudin,Real and Complex Analysis, 2nd ed. (McGraw-Hill, New York, 1974). · Zbl 0278.26001 |

[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). |

[24] | B. Hunt, T. Sauer and J. Yorke, Prevalence: A translation-invariant ?almost every? on infinite-dimensional spaces, preprint. · Zbl 0763.28009 |

[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). |

[27] | F. Takens, Detecting strange attractors in turbulence, inLecture Notes in Mathematics, No. 898 (Springer-Verlag, 1981). · Zbl 0513.58032 |

[28] | B. Townshend, Nonlinear prediction of speech signals, preprint. · Zbl 1056.92521 |

[29] | H. Whitney, Differentiable manifolds,Ann. Math. 37:645-680 (1936). · JFM 62.1454.01 |

[30] | J. Yorke, Periods of periodic solutions and the Lipschitz constant,Proc. Am. Math. Soc. 22:509-512 (1969). · Zbl 0184.12103 |

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