H. Abarbanel, R. Brown, and J. Kadtke, Prediction in chaotic nonlinear systems: Methods for time series with broadband Fourier spectra, preprint.
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
V. I. Arnold,Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, New York, 1983).
Google Scholar
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
D. S. Broomhead and G. P. King, Extracting qualitative dynamics from experimental data,Physics
20D:217–236 (1986).
Google Scholar
M. Casdagli, Nonlinear prediction of chaotic time series,Physica
35D:335–356 (1989).
Google Scholar
M. Casdagli, S. Eubank, D. Farmer, and J. Gibson, State-space reconstruction in the presence of noise, preprint.
W. Ditto, S. Rauseo, and M. Spano, Experimental control of chaos,Phys. Rev. Lett.
65:3211–3214 (1990).
Google Scholar
J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,Rev. Mod. Phys.
57:617–656 (1985).
Google Scholar
A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Hölder continuity for the inverse of Mañe's projection,Comptes Rendus, to appear.
K. Falconer,Fractal Geometry (Wiley, New York, 1990).
Google Scholar
J. D. Farmer and J. Sidorowich, Predicting chaotic time series,Phys. Rev. Lett.
59:845–848 (1987).
Google Scholar
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).
G. Golub and C. Van Loan,Matrix Computations, 2nd ed. (Johns Hopkins University Press, Baltimore, Maryland, 1989).
Google Scholar
E. Kostelich and J. Yorke, Noise reduction: Finding the simplest dynamical system consistent with the data,Physica
41D:183–196 (1990).
Google Scholar
E. Kostelich and J. Yorke, Noise reduction in dynamical systems,Phys. Rev. A
38:1649–1652 (1988).
Google Scholar
R. Mañé, On the dimension of the compact invariant sets of certain nonlinear maps, inLecture Notes in Mathematics, No. 898 (Springer-Verlag, 1981).
P. Marteau and H. Abarbanel, Noise reduction in chaotic time series using scaled probabilistic methods, preprint.
P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes,Ann. Acad. Sci. Fenn. Math.
1:227–224 (1975).
Google Scholar
F. Mitschke, M. Möller, and W. Lange, Measuring filtered chaotic signals,Phys. Rev. A
37:4518–4521 (1988).
Google Scholar
N. Packard, J. Crutchfield, D. Farmer, and R. Shaw, Geometry from a time series,Phys. Rev. Lett.
45:712 (1980).
Google Scholar
W. Rudin,Real and Complex Analysis, 2nd ed. (McGraw-Hill, New York, 1974).
Google Scholar
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
B. Hunt, T. Sauer and J. Yorke, Prevalence: A translation-invariant “almost every” on infinite-dimensional spaces, preprint.
T. Sauer and J. Yorke, Statistically self-similar sets, preprint.
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
F. Takens, Detecting strange attractors in turbulence, inLecture Notes in Mathematics, No. 898 (Springer-Verlag, 1981).
B. Townshend, Nonlinear prediction of speech signals, preprint.
H. Whitney, Differentiable manifolds,Ann. Math.
37:645–680 (1936).
Google Scholar
J. Yorke, Periods of periodic solutions and the Lipschitz constant,Proc. Am. Math. Soc.
22:509–512 (1969).
Google Scholar