# Attractor reconstruction for very short time series

Takens's theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence $\{s_j\}_{j=1}^N$ of observations of the state of a dynamical system. In order to obtain a correct attractor reconstruction, is there any bound on the length $N$ of that sequence? I think there is. Actually, the vector sequence $$\left(s_i,s_{i+\tau},s_{i+2\tau},\dots,s_{i+(m-1)\tau}\right)$$ describes the reconstructed attractor, where $\tau$ is an appropriate time delay and $m$ is the embedding dimension, and if $N=10$, $m=4$ and $\tau=3$ previous equation makes sense only for $i=1$.

## 1 Answer

You don’t do an attractor reconstruction for its own sake, so this really depends on what you want to do with it. Some examples:

• Suppose you already know what the system’s reconstructed attractor looks like, e.g., from another time series recorded from the same system. Further suppose, you want to know what state the system is currently in, e.g., for purposes of predicting or controlling the dynamics. Finally suppose that noise is negligible. In that case, a single point in your reconstructed phase space suffices to do what you want to do, and further points do not add any value. So, all you need is $m$ recordings with a distance of $τ$. (Arguably this is not a literal attractor reconstruction, but the idea should become clear.)

• The basic shape of a limit cycle is captured by three points in phase space, which require four time points from your time series. Of course, more data points would be better and allow you to more accurately capture what is going on, but if you are looking for a bare minimum, this would be one (for this specific case). Also, this amount of data does not suffice for being sure that your dynamics is a limit cycle and not, e.g., chaotic, but then again, no amount of data points suffices (in particular in the presence of noise).

• Fully capturing all the geometric details of a chaotic attractor requires infinitely many data points since you have a fractal attractor with arbitrarily fine detail.

• In case of an unknown dynamics, you already need a lot of data points to make a reasonable choice of $m$ and $τ$. Also, any further analysis you perform on the attractor, e.g., calculating Lyapunov exponents may require a considerable amount of data points. Finally, consider dynamical systems exhibiting rare excursions (extreme events), which can be arbitrarily rare; here you need to capture at least one excursion before your attractor reconstruction becomes useful with respect to this aspect of the dynamics. However, all these limitations are not imposed by the reconstruction process itself – they come from other steps in the pipeline surrounding it.