About: Takens's theorem

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In the study of dynamical systems, a delay embedding theorem gives the conditions under which a 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 (i.e., diffeomorphisms), but it does not preserve the geometric shape of structures in phase space. That is, there is a diffeomorphism that maps into such that the derivative of has full rank. is an embedding of the strange attractor in .

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rdfs:label	Takens's theorem (en)
rdfs:comment	In the study of dynamical systems, a delay embedding theorem gives the conditions under which a 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 (i.e., diffeomorphisms), but it does not preserve the geometric shape of structures in phase space. That is, there is a diffeomorphism that maps into such that the derivative of has full rank. is an embedding of the strange attractor in . (en)
dcterms:subject	Theorems in dynamical systems
Wikipage page ID	744335 (xsd:integer)
Wikipage revision ID	1081274695 (xsd:integer)
Link from a Wikipage to another Wikipage	Derivative Dynamical system Strange attractor Embedding Whitney embedding theorem James P. Crutchfield Norman Packard Euclidean space Floris Takens Diffeomorphism Rank (linear algebra) Attractor James A. Yorke Baire space Dynamical system (definition) Dynamical systems Chaos theory George Sugihara Theorems in dynamical systems Manifold Robert Shaw (physicist) Nonlinear dimensionality reduction Ricardo Mañé Whitney's embedding theorem L.-S. Young James Doyne Farmer R.M. May Smooth map Box counting dimension dbr:Martin_Casdagli dbr:Tim_Sauer
Link from a Wikipage to an external page	http://www.scholarpedia.org/article/Attractor_Reconstruction https://web.archive.org/web/20130917012451/http:/www.scientio.com/Products/ChaosKit
sameAs	Takens's theorem Takens's theorem
dbp:wikiPageUsesTemplate	dbt:Cite_conference dbt:Cite_journal dbt:More_footnotes dbt:Reflist
has abstract	In the study of dynamical systems, a delay embedding theorem gives the conditions under which a 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 (i.e., diffeomorphisms), 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 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 fordiscrete-time dynamical systems.The state space of the dynamical system is a -dimensional manifold . The dynamics is given by a smooth map Assume that the dynamics has a strange attractor with box counting dimension . Using ideas from Whitney's embedding theorem, can be embedded in -dimensional Euclidean space with That is, there is a diffeomorphism that maps into such that the derivative of has full rank. A delay embedding theorem uses an observation function to construct the embedding function. An observation function must be twice-differentiable and associate a real number to any point of the attractor . It must also be typical, so its derivative is of full rank and has no special symmetries in its components. The delay embedding theorem states that the function is an embedding of the strange attractor in . (en)
prov:wasDerivedFrom	wikipedia-en:Takens's_theorem?oldid=1081274695&ns=0
page length (characters) of wiki page	8164 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Takens's_theorem
is Link from a Wikipage to another Wikipage of	Empirical dynamic modeling Takens' theorem Whitney embedding theorem Floris Takens Taken's theorem List of theorems Nonlinear dimensionality reduction Delay coordinate embedding Delay embedding theorem
is Wikipage redirect of	Takens' theorem Taken's theorem Delay coordinate embedding Delay embedding theorem
is known for of	Floris Takens
is known for of	Floris Takens
is foaf:primaryTopic of	wikipedia-en:Takens's_theorem

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