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In the last segment, I showed you how
return maps work,

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how to go back and forth between
them and the time domain,

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and how they help you understand the
dynamics, as well as

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how to understand bifurcations in the
dynamics as the parameter value changes.

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I finished up with a 3rd representation,
the bifurcation diagram.

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Here's a bifurcation diagram of
the logistic map.

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On the vertical axis is a set of iterates
of the logistic map,

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at some parameter value R, which is
graphed on the horizontal axis.

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Just to remind you of the correspondence
between this kind of plot

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and the time domain, and the return map,
I'm gonna draw a few pictures.

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Here's a time domain plot of an orbit of
the logistic map

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at a low value of the parameter R
that is converging to a fixed point.

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On a return map, this orbit would look
like this.

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To construct a bifurcation diagram,
you remove the transient;

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that is, you iterate a bunch of times,
and throw those points away,

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and then you iterate a bunch more times,
and you plot those points

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as if you were looking at the top plot
edgeon from the side.

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In this case, those points would all fall
on top of each other, there.

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So again, each vertical slice of the
bifrucation diagram

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is one timedomain plot like this, with
the transient removed,

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viewed from the side.

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If we turn R up a little bit, the time
domain plot will look like this,

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the return map will look like this,

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and the point on the bifurcation diagram
will look like 2 dots.

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Again, three different representations
bring out three different things:

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the timedomain plot on the top left

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brings out the overall behavior of
the iterates;

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the return map on the lower left
brings out the geometry of why

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the iterates go where they go, & also the
correlation between successive iterates;

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the bifurcation plot brings out what
changes about the asymptotic behavior

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of the trajectory as R changes,
including bifurcations.

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Now if you repeat the procedure that we
just went through at a much finer grain,

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but using a computer instead of a tablet
and a stylus, what you'll see is this.

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There's actually one more step in there
which we'll circle back to

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at the end of this segment.

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Now, you can see all sorts of structure
in this plot.

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That's the main focus of this segment.

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First of all, you see the fixed point
coming along for low R, here,

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and then bifurcating into a 2cycle
right here,

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bifurcating into a 4cycle right here,
and then eventually,

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getting into a chaotic regime. That's
what this gray banded behavior is.

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That's what this righthand plot would
look like if you looked at it edgeon

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from the righthand side of the screen.

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Within the chaotic regimes, you also see
these "veils":

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areas where the attractor is darker
than in other areas.

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Those veils are related to what are called
"unstable periodic orbits",

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and we'll talk more about them later.

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As we've seen, there's this bifurcation
sequence

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from a fixed point, to a 2cycle, to a 4
cycle, to an 8cycle, & so on & so forth.

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That's called a "perioddoubling cascade"
for the obvious reason.

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I also showed you in the last segment
that there were regions of order

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within the chaos; that is, for some
Rvalue, there was chaos,

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but then if you raised R a little bit, you
went back into a periodic regime.

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This particular periodic regime starts out
with a 3cycle, and then goes to a 6cycle,

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and a 12cycle, and so on and so forth.

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So it's another periodic doubling
bifurcation sequence.

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You may remember, in the very first
segment of this course,

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I showed you the title page of a paper
called "Period3 Implies Chaos".

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The fact there is a period3 orbit
in this map is very, very significant.

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And if people are interested in that, I
can record an auxiliary video about that.

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Another interesting thing to note about
this structure

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is that it contains small copies of itself.

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If you were to zoom in on that piece of
the structure inside the red circle,

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it would look like the whole structure.

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That is, this is a fractal object.

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I'm sure many of you have hear about fractals.

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Fractals are sets that have noninteger
Hausdorff dimension

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(mathematically, that's the formal term).

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Informally, they're "selfsimilar".

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The second row of images here show
something called the Koch curve.

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The way you construct this triangle is by
taking an equilateral triangle,

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and then taking 3 equilateral triangles,
1/3 the size in the sense of edgelength,

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and sticking them to each exposed face
of that thing.

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Then you iterate; you take little
triangles

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and stick them to the sides of each of
those pointy faces, and keep going.

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Eventually, you'll get this beautiful
structure that looks alot like a snowflake

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Fractals play an interesting role
in mathematics.

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There are lots of examples of fractals
and fractallike structures in nature.

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Here's an example.

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Fractals are also useful analogs
for nature in computer graphics.

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Here's a beautiful fractal called
the Mandelbrot set,

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and this video is showing you that if you
zoom in on the Mandelbrot set,

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you keep seeing more and more structure;

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in fact, you keep seeing structure that is
selfsimilar.

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There's a whole new Mandelbrot set
way down in the tendrils of the old one.

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And you can keep zooming in
and zooming in,

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and you'll keep seeing
selfsimilar structure.

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I've included a link to that video on the
supplementary materials section

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of the Complexity Explorer website
for this course,

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right here, under the section for this
segment of this unit.

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Remember, this is where you should go

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for links to materials that you might need
to do the homework,

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like this Logistic Map app,

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for materials like this paper, which you
would look at

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if you wanted to learn more