James C Robinson 2005 Nonlinearity 18 2135 doi:10.1088/0951-7715/18/5/013
A topological delay embedding theorem for infinite-dimensional dynamical systems
James C Robinson
Show affiliationsRecommended by M J Field
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 
be a subset of a Hilbert space H with upper box-counting dimension 
and 'thickness exponent' τ, which is invariant under a Lipschitz map Φ. Take an integer k > (2 + τ)d, and suppose that 
, the set of all p-periodic points of Φ, satisfies 
for all p = 1, ..., k. Then a prevalent set of Lipschitz observation functions 
make the k-fold observation map ![\[
\begin{equation*}u\mapsto[h(u),h(\Phi(u)),h(\Phi^{k-1}(u))],\end{equation*}
\]](http://ej.iop.org/images/0951-7715/18/5/013/non186082ude001.gif)
one-to-one between and its image. The same result is true if is a subset of a Banach space provided that k > 2(1 + τ)d and 
.
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).
- PACS
- MSC
-
37Lxx Infinite-dimensional dissipative dynamical systems (See also 35Bxx, 35Qxx)
76F20 Dynamical systems approach to turbulence (See also 37-XX)
- Subjects
- Dates
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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
James C Robinson 2005 Nonlinearity 18 2135
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