Editing Takens's theorem

{{short description|Conditions under which a chaotic system can be reconstructed by observation}}
{{More footnotes needed|date=September 2020}}
[[File:Rössler attractor reconstructed by Taken's theorem, using different delay lengths..gif|thumb|[[Rössler attractor]] reconstructed by Takens' theorem, using different delay lengths. Orbits around the attractor have a period between 5.2 and 6.2.]]

In the study of [[dynamical systems]], 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 that system. The reconstruction preserves the properties of the dynamical system that do not change under smooth [[Change of basis|coordinate changes]] (i.e., [[diffeomorphism]]s), but it does not preserve the [[geometric shape]] of structures in [[phase space]].

'''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.

It is the most commonly used method for '''attractor reconstruction'''.<ref>{{Cite journal |last=Sauer |first=Timothy D. |date=2006-10-24 |title=Attractor reconstruction |journal=Scholarpedia |language=en |volume=1 |issue=10 |pages=1727 |doi=10.4249/scholarpedia.1727 |issn=1941-6016 |doi-access=free |bibcode=2006SchpJ...1.1727S }}</ref>

Delay embedding theorems are simpler to state for
[[Dynamical system (definition)|discrete-time dynamical systems]].
The state space of the dynamical system is a {{mvar|&nu;}}-dimensional [[manifold]] {{mvar|M}}. The dynamics is given by a [[smooth map]]

:<math>f: M \to M.</math>

Assume that the dynamics {{mvar|f}} has a [[strange attractor]] <math>A\sub M</math> with [[box counting dimension]] {{mvar|d{{sub|A}}}}.  Using ideas from [[Whitney's embedding theorem]], {{mvar|A}} can be embedded in {{mvar|k}}-dimensional [[Euclidean space]] with

:<math>k > 2 d_A.</math>

That is, there is a [[diffeomorphism]] {{mvar|&phi;}} that maps {{mvar|A}} into <math>\R^k</math> such that the [[derivative]] of {{mvar|&phi;}} has full [[rank (linear algebra)|rank]].

A delay embedding theorem uses an ''observation function'' to construct the embedding function.  An observation function <math>\alpha : M \to \R</math> must be twice-differentiable and associate a real number to any point of the attractor {{mvar|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>\varphi_T(x) = \bigl(\alpha(x), \, \alpha(f(x)), \, \dots, \, \alpha(f^{k-1}(x)) \, \bigr)</math>

is an [[Embedding#Differential topology|embedding]] of the strange attractor {{mvar|A}} in <math>\R^k.</math>

==Simplified version==

Suppose the <math>d</math>-dimensional 
state vector <math>x_t</math> evolves according to an unknown but continuous
and (crucially) deterministic dynamic. Suppose, too, that the
one-dimensional observable <math>y</math> is a smooth function of <math>x</math>, and “coupled”
to all the components of <math>x</math>. Now at any time we can look not just at
the present measurement <math>y(t)</math>, but also at observations made at times
removed from us by multiples of some lag <math>\tau: y_{t+\tau}, y_{t+2\tau} </math>, etc. If we use
<math>k</math> lags, we have a <math>k</math>-dimensional vector. One might expect that, as the
number of lags is increased, the motion in the lagged space will become
more and more predictable, and perhaps in the limit <math> k \to \infty </math> would become
deterministic. In fact, the dynamics of the lagged vectors become
deterministic at a finite dimension; not only that, but the deterministic
dynamics are completely equivalent to those of the original state space (precisely, they are related by a smooth, invertible change of coordinates,
or diffeomorphism). In fact, the theorem says that determinism appears once you reach dimension <math>2d+1</math>, and the minimal ''embedding dimension'' is often less.<ref>{{cite book|last1=Shalizi|first1=Cosma R.|editor1-last=Deisboeck|editor1-first=ThomasS|editor2-last=Kresh|editor2-first=J.Yasha|title=Complex Systems Science in Biomedicine|url=https://archive.org/details/complexsystemssc00kres|url-access=limited|date=2006|publisher=Springer US|isbn=978-0-387-30241-6|pages=[https://archive.org/details/complexsystemssc00kres/page/n47 33]–114|chapter=Methods and Techniques of Complex Systems Science: An Overview|doi=10.1007/978-0-387-33532-2_2|series=Topics in Biomedical Engineering International Book Series|arxiv=nlin/0307015|s2cid=11972113}}</ref><ref>{{Cite journal |last1=Barański |first1=Krzysztof |last2=Gutman |first2=Yonatan |last3=Śpiewak |first3=Adam |date=2020-09-01 |title=A probabilistic Takens theorem |url=https://iopscience.iop.org/article/10.1088/1361-6544/ab8fb8 |journal=Nonlinearity |volume=33 |issue=9 |pages=4940–4966 |doi=10.1088/1361-6544/ab8fb8 |arxiv=1811.05959 |bibcode=2020Nonli..33.4940B |s2cid=119137065 |issn=0951-7715}}</ref>

== Choice of delay ==
Takens' theorem is usually used to reconstruct strange attractors out of experimental data, for which there is contamination by noise. As such, the choice of delay time becomes important. Whereas for data without noise, any choice of delay is valid, for noisy data, the attractor would be destroyed by noise for delays chosen badly.

The optimal delay is typically around one-tenth to one-half the mean orbital period around the attractor.<ref>{{Cite book |last=Strogatz |first=Steven |title=Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering |date=2015 |isbn=978-0-8133-4910-7 |edition=Second |location=Boulder, CO |chapter=12.4 Chemical chaos and attractor reconstruction |oclc=842877119}}</ref><ref>{{Cite journal |last1=Fraser |first1=Andrew M. |last2=Swinney |first2=Harry L. |date=1986-02-01 |title=Independent coordinates for strange attractors from mutual information |url=https://link.aps.org/doi/10.1103/PhysRevA.33.1134 |url-access=subscription |journal=Physical Review A |volume=33 |issue=2 |pages=1134–1140 |doi=10.1103/PhysRevA.33.1134|pmid=9896728 |bibcode=1986PhRvA..33.1134F }}</ref>

== See also ==

* [[Whitney embedding theorem]]
* [[Nonlinear dimensionality reduction]]

==References==
{{Reflist}}

== 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 [[Robert Shaw (physicist)|R. Shaw]]
|volume = 45
|doi = 10.1103/PhysRevLett.45.712
|bibcode=1980PhRvL..45..712P
|issue = 9
}}
* {{cite conference
|book-title = 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
|book-title = 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 = [[Ricardo Mañé|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 = [[George Sugihara|G. Sugihara]] and [[R.M. May]]
|volume = 344
|doi = 10.1038/344734a0
|bibcode = 1990Natur.344..734S |s2cid = 4370167
}}
* {{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 = [[George Sugihara|G. Sugihara]]
|volume = 348
|doi = 10.1098/rsta.1994.0106
|bibcode = 1994RSPTA.348..477S
|issue = 1688 |s2cid = 121604829
}}
* {{cite journal
|journal = Science
|year = 1999
|title = Episodic fluctuations in larval supply
|pages = 1528&ndash;1530
|author = P.A. Dixon, M.J. Milicich, and [[George Sugihara|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 = [[George Sugihara|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 |doi-access = free
}}
* {{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 |s2cid = 2446456
}}
* {{cite journal
|journal = Remote Sensing of Environment
|year = 2015
|title = Estimating determinism rates to detect patterns in geospatial datasets
|pages = 11&ndash;20
|author = R. A. Rios, L. Parrott, H. Lange and R. F. de Mello
|volume = 156
|doi = 10.1016/j.rse.2014.09.019|bibcode = 2015RSEnv.156...11R
}}

==External links==
* [https://web.archive.org/web/20130917012451/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.
* [https://sugiharalab.github.io/EDM_Documentation/] Empirical Dynamic Modelling tools pyEDM and rEDM use embedding for analyses, prediction, and causal inference.

[[Category:Theorems in dynamical systems]]