We ended the last segment with the Takens theorem which concerns a delay coordinate
embedding of a time-series data set from a dynamical system.
Today's task is to dig into one of the really important pieces of this.
The words at the end of the second line and the beginning of the third:
"diffeomorphic to" and "have the same topology as."
Now you remember the goal of this procedure. The goal is to undo a
projection. The procedure itself we covered in the previous section.
You plot delayed versions of the measured quantity against itself.
If you have good data and you do the embedding correctly, the results are
guaranteed to be topologically identical to the true dynamics.
The "good data" part of what I said is the note at the bottom right on this slide.
The "do it right" part is the colored words.
We'll come back to all of that later in this unit.
Now since topology is not a prerequisite for this class, I'm going to do Topology
101 in two minutes.
Topology is the fundamental mathematics of shape, as I said last time.
What I mean by that: what the concept of shape becomes if you don't measure.
The word "geometry" has the word "metric" buried in it. A metric is a way
to measure something. The word "topology" does not have that. It's not "topo-metry".
Size does not matter. Since size doesn't matter, all of these are the same thing.
When you're talking about the topology of an object, only the number of pieces,
or the number of holes matters.
The two objects at the top here, the coffee cup and the donut,
have the same topology because they both have one hole and one piece.
Their geometry is very different however.
The two objects at the bottom have very similar geometries.
The one on the left, which is a colander which you use to drain pasta however,
has lots and lots of holes whereas the bowl on the right doesn't have any holes.
So the two objects on the bottom have similar geometry and different topology.
The two objects on the top have the same topology and very different geometry.
Now think about a donut and a coffee mug made out of clay or dough.
You could deform one into the other with- out destroying or creating pieces or
destroying or creating holes.
If you can do that, the two objects have the same topology.
And for those who know this object, obviously I'm referring to Betti numbers.
The mathematical form of transformations like that deforming of the dough that
don't make or break pieces or holes, is called a diffeomorphism.
A diffeomorphism is one to one, onto, invertible, and differentiable
in both directions.
And a correct embedding is related to the true dynamics by such a transformation
if the conditions of the theorem are met.
And what that means, is that the reconstructed dynamics here on the left
have the same topology as the true dynamics on the right.
The real and reconstructed attractors don't look alike to us because our eyes
respond to geometry, not topology. But they're identical in some very formal and
powerful mathematical ways. So why is all this useful?
You may remember my definition of bifurcation as "topological changes in
the attractor", for instance. Now you know what that means.
Topology really matters. Many of the important properties of
dynamical systems, such as the Lyapunov exponent, are invariant under those
transformations, the diffeomorphisms, that preserve topology. And all of that
means, that you can measure one thing from a very complex system, do the embedding,
compute the value of one of these dynamical invariants, and assert that your
answer holds for the true unobserved dynamics inside the black box.
Which is pretty darned amazing. Now taken to an extreme that means that I
could take a thermometer, stick it outside my window, measure a time series of the
temperatures outside my window and from that time series I could reconstruct the
dynamics of weather of the Western hemisphere. Now that does not work for a
couple of very important reasons having to do with how you would actually have
to do the embedding to get that to work and how much data you would need
to get that to work properly.
And that is what we'll talk about in the next segment.