In Dynamical systems theory, Takens' embedding theorem is as follows:

Suppose that a measured time series $y(1), y(2), \ldots, y(N)$ lies on a $D$-dimensional attractor of an $n$th-order deterministic dynamical system. The starting point obtains an embedding from the recorded data. A convenient, though not unique, representation is achieved by using delay coordinates, for which a delay vector has the following form: $$ \mathbf{y}(k)=\left[\begin{array}{llll} y(k) & y(k-\tau) & \cdots & y\left(k-\left(d_{\mathrm{e}}-1\right) \tau\right) \end{array}\right]^{\mathrm{T}}, $$ where $d_{\mathrm{e}}$ is the embedding dimension and $\tau$ is the delay time. Takens has shown that embeddings with $d_{\mathrm{e}}>2 n$ will be faithful generically so that there is a smooth map $f: \mathbb{R}^{d_{\mathrm{e}}} \mapsto \mathbb{R}$ such that $$ y(k+1)=f(\mathbf{y}(k)) $$ for all integers $k$, and where the forecasting time $T$ and $\tau$ are also assumed to be integers.

I want to know if it is always decidable to compute an appropriate $\tau$? By decidable, I mean the Turing machine model.

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    $\begingroup$ Is the question whether it's decidable if a given $\tau$ gives an embedding? Or whether one can always compute an appropriate $\tau$? Also it might be good to specify the model of computation (Turing machine, Blum-Shub-Smale real numbers, ...) $\endgroup$ Jan 13, 2022 at 19:33
  • $\begingroup$ @MartinM.W. Thanks for the comment. I've just edited my question. $\endgroup$ Jan 14, 2022 at 6:48


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