A topological delay embedding theorem

doi:10.1088/0951-7715/18/5/013

A topological delay embedding theorem for infinite-dimensional dynamical systems

James C Robinson 2005 Nonlinearity 18 2135-2143 doi: 10.1088/0951-7715/18/5/013

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James C Robinson
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
E-mail: jcr@maths.warwick.ac.uk

Recommended by M J Field

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 ${\mathcal{A}}$ be a subset of a Hilbert space H with upper box-counting dimension $d({\mathcal{A}})=d$ and 'thickness exponent' τ, which is invariant under a Lipschitz map Φ. Take an integer k > (2 + τ)d, and suppose that ${\mathcal{A}}_p$ , the set of all p-periodic points of Φ, satisfies $d({\mathcal{A}}_p)<p/(2+\tau)$ for all p = 1, ..., k. Then a prevalent set of Lipschitz observation functions $h:H\rightarrow {\mathbb{R}}$ make the k-fold observation map

$\begin{equation*}u\mapsto[h(u),h(\Phi(u)),h(\Phi^{k-1}(u))],\end{equation*}$

one-to-one between ${\mathcal{A}}$ and its image. The same result is true if ${\mathcal{A}}$ is a subset of a Banach space provided that k > 2(1 + τ)d and $d({\mathcal{A}}_p)<p/(2+2\tau)$ .

The result follows from a version of the Takens theorem for Hölder 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).

Mathematics Subject Classification: 37L30, 35B41, 35Q30, 76F20

Print publication: Issue 5 (September 2005)
Received 10 September 2004, in final form 27 April 2005
Published 1 July 2005

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A topological delay embedding theorem for infinite-dimensional dynamical systems

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