# Decidability of choosing delay in Takens' theorem

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.

• 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, ...) Jan 13, 2022 at 19:33
• @MartinM.W. Thanks for the comment. I've just edited my question. Jan 14, 2022 at 6:48