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A topological delay embedding theorem for infinitedimensional dynamical systems
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UNSPECIFIED (2005) A topological delay embedding theorem for infinitedimensional dynamical systems. NONLINEARITY, 18 (5). pp. 21352143. doi:10.1088/09517715/18/5/013
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Official URL: http://dx.doi.org/10.1088/09517715/18/5/013
Abstract
A time delay reconstruction theorem inspired by that of Takens (1981 Springer Lecture Notes in Mathematics vol 898, pp 36681) is shown to hold for finitedimensional subsets of infinitedimensional spaces, thereby generalizing previous results which were valid only for subsets of finitedimensional spaces.
Let A be a subset of a Hilbert space H with upper boxcounting dimension d(A) = d and 'thickness exponent' tau, which is invariant under a Lipschitz map Phi. Take an integer k > (2 + tau)d, and suppose that A(p), the set of all pperiodic points of Phi, satisfies d(A(p)) < p/(2 + tau) for all p = 1,..., k. Then a prevalent set of Lipschitz observation functions h : H > R make the kfold observation map u > [h(u), h(Phi(u)), h(Phi(k1) (u))], onetoone between A and its image. The same result is true if A is a subset of a Banach space provided that k > 2(1 + tau)d and d(A(p)) < p/(2 + 2 tau).
The result follows from a version of the Takens theorem for Holder continuous maps adapted from Sauer et al (1991 J. Stat. Phys. 65 52947), and makes use of an embedding theorem for finitedimensional sets due to Hunt and Kaloshin (1999 Nonlinearity 12 126375).
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics Q Science > QC Physics 

Journal or Publication Title:  NONLINEARITY  
Publisher:  IOP PUBLISHING LTD  
ISSN:  09517715  
Official Date:  September 2005  
Dates: 


Volume:  18  
Number:  5  
Number of Pages:  9  
Page Range:  pp. 21352143  
DOI:  10.1088/09517715/18/5/013  
Publication Status:  Published 
Data sourced from Thomson Reuters' Web of Knowledge
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