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In [[mathematics]], a '''delay embedding theorem''' gives the conditions under which a [[chaos theory|chaotic]] [[dynamical system]] can be reconstructed from a sequence of observations of the state of a dynamical system.   The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes, but it does not preserve the geometric shape of structures in phase space.
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'''Takens' theorem''' is the 1981 delay embedding theorem of [[Floris Takens]].  It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a [[Baire space|generic]] function.  Later results replaced the smooth attractor with a set of arbitrary [[box counting dimension]] and the class of generic functions with other classes of functions.
 
Delay embedding theorems are simpler to state for
[[Dynamical_system#Types of dynamical systems|discrete-time dynamical system]]s.
The state space of the dynamical system is a ν-dimensional manifold ''M''. The dynamics is given by a [[smooth map]]
 
:<math>f: M \to M.</math>
 
Assume that the dynamics ''f'' has a [[strange attractor]] ''A'' with [[box counting dimension]] ''d<sub>A</sub>''. Using ideas from [[Whitney's embedding theorem]], ''A'' can be embedded in ''k''-dimensional [[Euclidean space]] with
 
:<math>k > 2 d_A.\ </math>
 
That is, there is a [[diffeomorphism]] φ that maps ''A'' into '''R'''<sup>''k''</sup> such that the [[derivative]] of φ has full [[rank (linear algebra)|rank]].
 
A delay embedding theorem uses an ''observation function'' to construct the embedding function.  An observation function α must be twice-differentiable and associate a real number to any point of the attractor ''A''. It must also be [[Baire space|typical]], so its derivative is of full rank and has no special symmetries in its components. The delay embedding theorem states that the function
 
:<math>\phi_T(x)=\left(\alpha(x), \alpha\left(f(x)\right), \dots, \alpha\left(f^{k-1}(x)\right)\right)</math>
 
is an embedding of the strange attractor ''A''.
 
==References==
{{Unreferenced|date=May 2008}}
 
== Further reading ==
* {{cite journal
|journal = Physical Review Letters
|year = 1980
|title = Geometry from a time series
|pages = 712&ndash;716
|author = [[Norman Packard|N. Packard]], [[James P. Crutchfield|J. Crutchfield]], [[James Doyne Farmer|D. Farmer]] and [[R. Shaw]]
|volume = 45
|doi = 10.1103/PhysRevLett.45.712
|bibcode=1980PhRvL..45..712P
|issue = 9
}}
* {{cite conference
|booktitle = Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898
|year = 1981
|title = Detecting strange attractors in turbulence
|pages = 366&ndash;381
|author = [[Floris Takens|F. Takens]]
|publisher = Springer-Verlag
|editor = [[D. A. Rand]] and [[L.-S. Young]]
}}
* {{cite conference
|booktitle = Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898
|year = 1981
|title = On the dimension of the compact invariant sets of certain nonlinear maps
|pages = 230&ndash;242
|author = R. Mañé
|publisher = Springer-Verlag
|editor = D. A. Rand and L.-S. Young
}}
* {{cite journal
|journal = Nature
|year = 1990
|title = Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series
|pages = 734&ndash;741
|pmid = 2330029
|issue = 6268
|author = [[G. Sugihara]] and [[R.M. May]]
|volume = 344
|doi = 10.1038/344734a0
|bibcode = 1990Natur.344..734S }}
* {{cite journal
|journal = Journal of Statistical Physics
|year = 1991
|title = Embedology
|pages = 579&ndash;616
|author = [[Tim Sauer]], [[James A. Yorke]], and [[Martin Casdagli]]
|volume = 65
|doi = 10.1007/BF01053745
|bibcode = 1991JSP....65..579S
|issue = 3–4 }}
* {{cite journal
|journal = Phil. Trans. R. Soc. Lond. A
|year = 1994
|title = Nonlinear forecasting for the classification of natural time series
|pages = 477&ndash;495
|author = [[G. Sugihara]]
|volume = 348
|doi = 10.1098/rsta.1994.0106
|bibcode = 1994RSPTA.348..477S
|issue = 1688 }}
* {{cite journal
|journal = Science
|year = 1999
|title = Episodic fluctuations in larval supply
|pages = 1528&ndash;1530
|author = [[P.A. Dixon]], [[M.J. Milicich]], and [[G. Sugihara]]
|volume = 283
|doi = 10.1126/science.283.5407.1528
|pmid=10066174
|bibcode = 1999Sci...283.1528D
|issue=5407}}
* {{cite journal
|journal = PNAS
|year = 1999
|title = Residual delay maps unveil global patterns of atmospheric nonlinearity and produce improved local forecasts
|pages = 210&ndash;215
|author = [[G. Sugihara]], [[M. Casdagli]], [[E. Habjan]], [[D. Hess]], [[P. Dixon]] and [[G. Holland]]
|volume = 96
|pmid=10588685
|issue = 25
|pmc = 24416
|doi=10.1073/pnas.96.25.14210
|bibcode = 1999PNAS...9614210S }}
* {{cite journal
|journal = Nature
|year = 2005
|title = Distinguishing random environmental fluctuations from ecological catastrophes for the North Pacific Ocean
|first4 = G
|last4 = Sugihara
|first3 = AJ
|last3 = Lucas
|first2 = SM
|pages = 336&ndash;340
|last2 = Glaser
|pmid = 15902256
|issue = 7040
|author = [[C. Hsieh]]
|volume = 435
|doi = 10.1038/nature03553
|bibcode = 2005Natur.435..336H }}
 
==External links==
* [http://www.scholarpedia.org/article/Attractor_Reconstruction Attractor Reconstruction (scholarpedia)]
* [http://www.scientio.com/Products/ChaosKit] Scientio's ChaosKit product uses embedding to create analyses and predictions. Access is provided online via a web service and graphic interface.
 
[[Category:Theorems in dynamical systems]]

Latest revision as of 14:32, 25 September 2014

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