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So let's summarize Unit 5.
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This was the second of two units on bifurcations
and bifurcation diagrams.
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This unit focus exclusively on the logistic
map, the iterated logistic function.
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I began by reminding us of the idea of final
state diagrams.
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So for the logistic equation, for a given r value,
you might make a final state diagram as follows.
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Iterate the function for a hundred times,
then iterate another two hundred times and
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plot those iterates on a unit interval, and
the result is a final state diagram and this
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is very similar to the phase line for differential
equations.
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It shows the equilibrium or long-term behavior
but it doesn't show arrows for the directions the
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orbits move, and that's because for the logistic
equation and iterated functions in general, the
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orbit can bounce around and it doesn't move
smoothly through all continuous values.
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So here are some examples, and we did a bunch
early on in this unit.
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Here's a value of r which has period two behavior,
it hits that period two behavior very quickly.
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The final state diagram had two dots, these
two final values, so I can iterate out for a hundred
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and then iterate the next two hundred and it would
just be bouncing back and forth between these two.
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It's a little bit more interesting for cases
where we have aperiodic behavior.
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Here is a different r value that has an aperiodic
orbit, and here the final state diagram would
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consist of many many points between these
two extreme values.
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So it looks like the orbits never get much below
0.18 or above 0.96, but they can take on all values
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in between and so if I plotted two hundred
orbits, two hundred iterates, I would see
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two hundred dots and they would essentially
fill up this line.
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So those are final state diagrams and then
to make a bifurcation diagram, we just glue
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all of those final state diagrams together.
So each vertical slice of a bifurcation diagram
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is a single final state diagram.
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Another way I'd like to think about this is
that the bifurcation diagram is like a dictionary
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that lets you look up the behavior for
different r values.
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So suppose someone says, "what does a logistic
equation do for r=3.6?" Say, well, I don't know
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but I can look that up in the bifurcation diagram.
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It's a dictionary or a compendium of all
possible behaviors for all different r values.
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So I look over here at 3.6, and I would go
up, and imagine taking a really thin slice
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out of this, and that would be my final state
diagram for that value.
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So the bifurcation diagram summarizes a
great deal of information.
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And we noticed that there's some interesting and
intriguing, I hope, patterns in this bifurcation diagram.
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There are lots of transitions from periodic
behavior to chaotic behavior, periodic to chaotic,
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periodic to chaotic, and so on. So there's a
lot going on in this bifurcation diagram.
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To get a closer look, I mentioned or introduced
to you to a web program that lets you zoom in
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on the bifurcation diagram many many many
times, and explore, see what you see, see
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what it looks like. So we zoomed in and explored,
and along the way as we were doing that, we needed
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to adjust some times the number of orbits
that were skipped and we need to do that
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if there's long transient behavior, takes a
long time to reach those final states.
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And sometimes we need to increase the number
of orbits plotted. If we zoomed in a lot in the
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vertical direction, so that we're only seeing
a small region, we lose resolution, so by
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plotting more points we can gain resolution.
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The plotting of points on the screen takes
a long time, so don't let this be too large,
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unless you're zoomed in a fair amount.
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So let me close this summary just by reminding
you that the bifurcation diagram, this amazingly
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complex figure all comes from this simple
equation: f(x) = rx(1-x). And this is just a parabola.
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A parabola is almost as simple a function as
you could come up with, it's not a complicated
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topic from algebra, but when iterated, there's
an almost infinite amount of, if not complexity,
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then certainly an infinite amount of structure or
regularity in these fractal patterns that one sees.
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So the function is simple and the iteration
process is also very simple.
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The code that I used to make the bifurcation
diagram is a short piece of code, in Python,
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the one I used to make the figure a couple
of slides ago, that code is just forty-seven
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lines long and that includes comments on all
the plotting commands as well.
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The basic algorithm, the basic recipe for
iteration is a very very simple one.
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So iteration, this very simple and infinitely
repetitive process, and a very simple function
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like a parabola, can produce this remarkably
complex structure that we see in the bifurcation
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diagram.
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It's pretty amazing to me that such a complex
structure such as the bifurcation diagram can
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arise from such a simple equation like the
logistic equation.
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If you have experienced programming, I definitely suggest
that you try coding up a bifurcation diagram program.
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it's fun to do, it's not a hard programming task, and
it's very satisfying, and I think sort of amazing, to see
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the bifurcation diagram emerge as a result
of the program you wrote.
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When I write such a program like I did for this
unit, when I see the bifurcation diagram emerge
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I sometimes sort of wonder, have this question,
where did that come from, who did that?
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Well, of course, I know I did it, I wrote the
program, and maybe the computer did it,
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it's making the picture. But where does the
infinite structure of the bifurcation diagram
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come from? Where does it arise? For that
matter where does the randomness, unpredictability
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and butterfly effect arise out of this simple
parabolic equation.
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I think the answer in both of these cases
is that it arises from iteration, that the act
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of iterating, itself a very simple process
and infinitely repetitious, the act of iterating
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a simple equation gives rise to properties
or features that weren't present in the equation
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in the first place.
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We've seen randomness and unpredictability
arise, and we've seen a really rich complexity
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arise from a very simple equation that gets iterated.
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This in turn calls attention to one of the
things that I think is different and maybe
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noteworthy about the study of dynamical systems.
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In physics, and probably elsewhere in the
physical sciences, sometimes when one has
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the equation in hand, the story is sort of
over, at least as far as the science is concerned.
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That we try to understand a process, and once
we have an equation for it, the process is
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understood, story over.
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One can then use that equation to make
predictions build bridges, to build computers.
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But the understanding is encoded in the equation itself.
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I think the same isn't true for the iterated
logistic equation.
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If you have the equation, it's just the beginning.
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The equation itself, just looking at the equation,
doesn't make evident that there's a butterfly effect
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lurking, or that there's this infinite complexity
and structure in the bifurcation diagram.
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Those are features of the system that only
emerge via the act of iteration.
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And, very often, the only way to see what the
system will do is to see what the system will do.
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And what I mean by that is one has to just
start iterating and see what happens.
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There's not always an analytic method, a
pencil and paper, a deductive way to figure
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out if an iterated function will be chaotic or not.
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But if you iterate it on a computer, you can
see very quickly what's going on.
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So this is a feature of dynamical systems
that I think is somewhat different than how
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equations are used elsewhere. The equation
maybe isn't the end of the story, but it's
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the beginning. One has to iterate it and take
a bit of a more experimental approach, to see
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what properties emerge from the act of iteration.
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And often the properties that emerge are properties
that weren't present in the original equation.
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So, in any event, this brings us to the end of Unit 5.
We've looked at the bifurcation diagram for the
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logistic equation.
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In the next unit, we'll focus in on this feature
of repeated period doublings, those pitchforks
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of sideways U's that come up again and again
and again in the bifurcation diagram.
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And amazingly, we'll see that this isn't just
some mathematical or geometric curiosity,
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but it actually has some really important
statements to make about actual physical
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properties of systems in the material world.
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See you then.