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Takens' theorem, the way I understand it in simple terms, states that, under specific conditions, you can roughly reconstruct the state-space of a dynamical system by delay-embedding only one of its time-series projections.

I'm trying to understand intuitively how this is possible.

The way I was looking at it was: let's take the Lorenz attractor. You can reconstruct the dynamics of that system by delay-embedding only time series projection of the X axis. Imagine the new coordinates will be: $X, X(t - l), X(t - 2l)$, where l is the delay. This was my logic: this works because X contains redundant information about $Y$ and $Z$. That means $X(t - l)$ is kind of like $Y$ and $X(t-2l)$ is a kind of like $Z$.

Is my logic sound? I would like to know if there's any established intuition behind why this is possible.

Vivek Thazhathattil
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Nathan
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Intuitively, the "hidden" dimensions are still coupled to the "visible" dimension, their values influence where it goes next, how "curved" the graph of the visible component is etc. This deterministic influence allows to reconstruct the hidden part of the dynamic from the time series of the visible component.

For a general discussion of the theory and how Takens theorem applies to invariant sets of fractal dimension, see scholarpedia: Attractor reconstruction. A condensed version focused on the Lorenz system can be found in Understanding Takens' Embedding theorem.

As to the Lorenz system specifically I discussed how $y$ and z can be more-or-less reconstructed from $x$ with 2 time delays in Why Lorenz attractor can be embedded by a 3-step time delay map?. This obviously is much less than the embedding dimension 7 of the original Takens theorem and 5 of the fractal version.

In Lorenz Equations, embedding and Takens’ theorem was discussed why the reconstruction is not possible starting from the $z$ values due to the symmetry $(x,y,z)\to (-x,-y,z)$ of the system.

Another way the theorem can "fail" is if you take $2$ independent copies of the Lorenz system with different initial values. Then the $x$ component of the first system will contain no information of the second system. But actually, that this situation is to be excluded in the theorem is included in the "generic" assumption, in that a thin set of signals may have to be excluded from the statement. Conversely, any generic linear combination of both $x$ components should be a valid signal for the combined system. As the combined box-counting dimension should be slightly larger than $4$, an embedding dimension of $9$ (or possibly smaller) would be sufficient.

Lutz Lehmann
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  • Thank you very much for your answer and the useful resources. That first paragraph makes sense to me, and it's in line with how I was thinking about the problem. One related question if I may ask: intuitively what's the significance of time-delaying? Does it have anything to do with "bringing out" that deterministic influence of the hidden dimensions? What happens if we don't time-delay and keep using the same values for all coordinates of the reconstructed dynamic? – Nathan Sep 10 '20 at 11:23
  • I'm not sure what you mean with "keep using the same values for all coordinates". Your dynamical system is a black box for this theorem, you only have one measurable signal output. So the only available information to make conclusions about the internal complexities of the black box is the variation of that signal over time, so for a finite amount of data you get a (moving) slice of the time series, that is, values from a sequence of time delays. – Lutz Lehmann Sep 10 '20 at 11:34
  • Makes total sense. Thank you. – Nathan Sep 10 '20 at 11:47