Intuition behind Takens' Theorem Takens' theorem, the way I understand it in simples 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 of the fact that X contains redundant information about $Y$ and $Z$. That means $X(t - l)$ is kind of like $Y$ and $X(t-l)$ 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.
 A: 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.
