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