When is an attractor meaningful?

I’m originally a computer scientist; so I hope my question is not trivial. I’m working with time series and want to reconstruct the phase space from the time series based on time-lagged versions of time series. For this purpose I need to calculate $m$ and $\tau$ which are the embedding dimension and time-lag respectively.

But: Today I was seeing some experiments based on some random $m$ and $τ$ and I saw some topology in their attractor after which this question came to my mind:

Regardless of how one chooses $m$ and $\tau$, if the attractor has a meaningful topology (i.e. following a geometrically structured shape), does it mean that this attractor could capture some meaningful dynamics?

In other words assume that I calculated $m=M$ and $\tau=T$, but I get a meaningful attractor shape for another $m$ and $\tau$. How can I interpret it?

Let’s look at what happens when you make non-optimal selections of $τ$ and $m$:

• If $m$ is too low, the attractor will not be completely unfolded, i.e., parts of your reconstructed attractor will overlap when they shouldn’t. This is equivalent to trajectories crossing in phase space, which means that what you have reconstructed is not a proper attractor or phase-space reconstruction, in particular it’s not physically meaningful.

Example: Every two-dimensional picture of the Lorenz attractor, e.g., this one.

• If $m$ is too high, nothing bad happens except that you have spurious dimensions to deal with when further processing your result.

Example: You embed the attractor of a driven and damped oscillator (a circle) in three dimensions, instead of two. It is still a circle.

• If $τ$ is slightly off the optimum, your reconstruction will not be optimal for seeing the structure, but it will still be a valid reconstruction. In fact, Takens’ embedding theorem gives you that almost every choice of $τ$ yields a proper embedding, if you only go by topological equivalence. The advantage of the optimal $τ$ rather are that you can best see the structure of the attractor, that numerical analyses of the attractor are more feasible, and that measurement noise and similar have a minimum impact on your reconstruction.

Example: If you use a non-optimal $τ$ for embedding the attractor of a driven and damped oscillator, you do not get a circle, but an ellipsis. The further off your $τ$ is, the flatter is the ellipsis. It is easy to see that if your ellipsis is sufficiently flat and you have sufficiently strong measurement noise, the structure is lost. Furthermore there are discrete choices of $τ$ where the ellipsis collapses to a line, which are the reason why only almost every choice of $τ$ yields a proper embedding.

• If $τ$ is far off the optimum, you still are very likely to have a proper embedding in the topological sense but the attractor’s structure is unnecessarily complicated from a visual point of view and further analysis of your results become more difficult.

Example: Fraser and Swinney, PRA 33, 1134 (1986) contains some illustrative examples, in particular in Fig. 1.

So, if you choose $m$ and $τ$ non-optimal, you can still get meaningful results. Keep in mind, that attractor reconstruction can only unveil the topology of the attractor, and topologically equivalent shapes can be very different by other criteria. Finally, note that there is no straightforward way to obtain an optimal $τ$ and $m$ for empirical data – you cannot know whether your choices are good before you actually looked at the results (and even then, it may be difficult to tell).

• Thanks for your comprehensive answer :) I'll wait a bit more and if no one else provided a better answer I'll accept yours. Thank you again! – Kasra Manshaei Mar 2 '16 at 8:25

I don't believe the time lag is critical. Selecting a bad one might mean that you need to analyze more data to fill in the phase space, but it should still generate it given sufficient time. As for the meaningful aspect, that does depend critically on the embedding dimension. Usually one uses the human capacity for pattern recognition to extract meaning from such data. If the imbedding dimension is poorly selected, you are only seeing a slice of the phase space and that may complicate pattern recognition.

That there is any real meaning in this endeavor depends on the degree of universality in the class of nonlinear problem you are investigating.