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A topological delay embedding theorem for infinite-dimensional dynamical systems
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UNSPECIFIED (2005) A topological delay embedding theorem for infinite-dimensional dynamical systems. NONLINEARITY, 18 (5). pp. 2135-2143. doi:10.1088/0951-7715/18/5/013 ISSN 0951-7715.
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Official URL: http://dx.doi.org/10.1088/0951-7715/18/5/013
Abstract
A time delay reconstruction theorem inspired by that of Takens (1981 Springer Lecture Notes in Mathematics vol 898, pp 366-81) is shown to hold for finite-dimensional subsets of infinite-dimensional spaces, thereby generalizing previous results which were valid only for subsets of finite-dimensional spaces.
Let A be a subset of a Hilbert space H with upper box-counting 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 p-periodic 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 k-fold observation map u -> [h(u), h(Phi(u)), h(Phi(k-1) (u))], one-to-one 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. 65 529-47), and makes use of an embedding theorem for finite-dimensional sets due to Hunt and Kaloshin (1999 Nonlinearity 12 1263-75).
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Journal or Publication Title: | NONLINEARITY | ||||
Publisher: | IOP PUBLISHING LTD | ||||
ISSN: | 0951-7715 | ||||
Official Date: | September 2005 | ||||
Dates: |
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Volume: | 18 | ||||
Number: | 5 | ||||
Number of Pages: | 9 | ||||
Page Range: | pp. 2135-2143 | ||||
DOI: | 10.1088/0951-7715/18/5/013 | ||||
Publication Status: | Published |
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