So let's summarize Unit 5.

This was the second of two units on bifurcations
and bifurcation diagrams.

This unit focus exclusively on the logistic
map, the iterated logistic function.

I began by reminding us of the idea of final
state diagrams.

So for the logistic equation, for a given r value, 
you might make a final state diagram as follows.

Iterate the function for a hundred times,
then iterate another two hundred times and

plot those iterates on a unit interval, and
the result is a final state diagram and this

is very similar to the phase line for differential
equations.

It shows the equilibrium or long-term behavior
but it doesn't show arrows for the directions the

orbits move, and that's because for the logistic
equation and iterated functions in general, the

orbit can bounce around and it doesn't move
smoothly through all continuous values.

So here are some examples, and we did a bunch
early on in this unit.

Here's a value of r which has period two behavior,
it hits that period two behavior very quickly.

The final state diagram had two dots, these
two final values, so I can iterate out for a hundred

and then iterate the next two hundred and it would
just be bouncing back and forth between these two.

It's a little bit more interesting for cases
where we have aperiodic behavior.

Here is a different r value that has an aperiodic
orbit, and here the final state diagram would

consist of many many points between these
two extreme values.

So it looks like the orbits never get much below
0.18 or above 0.96, but they can take on all values

in between and so if I plotted two hundred
orbits, two hundred iterates, I would see

two hundred dots and they would essentially
fill up this line.

So those are final state diagrams and then
to make a bifurcation diagram, we just glue

all of those final state diagrams together.
So each vertical slice of a bifurcation diagram

is a single final state diagram.

Another way I'd like to think about this is 
that the bifurcation diagram is like a dictionary

that lets you look up the behavior for
different r values.

So suppose someone says, "what does a logistic
equation do for r=3.6?" Say, well, I don't know

but I can look that up in the bifurcation diagram.

It's a dictionary or a compendium of all
possible behaviors for all different r values.

So I look over here at 3.6, and I would go
up, and imagine taking a really thin slice

out of this, and that would be my final state
diagram for that value.

So the bifurcation diagram summarizes a
great deal of information.

And we noticed that there's some interesting and 
intriguing, I hope, patterns in this bifurcation diagram.

There are lots of transitions from periodic
behavior to chaotic behavior, periodic to chaotic,

periodic to chaotic, and so on. So there's a
lot going on in this bifurcation diagram.

To get a closer look, I mentioned or introduced
to you to a web program that lets you zoom in

on the bifurcation diagram many many many
times, and explore, see what you see, see

what it looks like. So we zoomed in and explored,
and along the way as we were doing that, we needed

to adjust some times the number of orbits
that were skipped and we need to do that

if there's long transient behavior, takes a
long time to reach those final states.

And sometimes we need to increase the number
of orbits plotted. If we zoomed in a lot in the

vertical direction, so that we're only seeing
a small region, we lose resolution, so by

plotting more points we can gain resolution.

The plotting of points on the screen takes
a long time, so don't let this be too large,

unless you're zoomed in a fair amount.

So let me close this summary just by reminding
you that the bifurcation diagram, this amazingly

complex figure all comes from this simple
equation: f(x) = rx(1-x). And this is just a parabola.

A parabola is almost as simple a function as
you could come up with, it's not a complicated

topic from algebra, but when iterated, there's
an almost infinite amount of, if not complexity,

then certainly an infinite amount of structure or 
regularity in these fractal patterns that one sees.

So the function is simple and the iteration
process is also very simple.

The code that I used to make the bifurcation
diagram is a short piece of code, in Python,

the one I used to make the figure a couple
of slides ago, that code is just forty-seven

lines long and that includes comments on all
the plotting commands as well.

The basic algorithm, the basic recipe for
iteration is a very very simple one.

So iteration, this very simple and infinitely
repetitive process, and a very simple function

like a parabola, can produce this remarkably
complex structure that we see in the bifurcation

diagram.

It's pretty amazing to me that such a complex
structure such as the bifurcation diagram can

arise from such a simple equation like the
logistic equation.

If you have experienced programming, I definitely suggest 
that you try coding up a bifurcation diagram program.

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

the bifurcation diagram emerge as a result
of the program you wrote.

When I write such a program like I did for this
unit, when I see the bifurcation diagram emerge

I sometimes sort of wonder, have this question,
where did that come from, who did that?

Well, of course, I know I did it, I wrote the
program, and maybe the computer did it,

it's making the picture. But where does the
infinite structure of the bifurcation diagram

come from? Where does it arise? For that
matter where does the randomness, unpredictability

and butterfly effect arise out of this simple
parabolic equation.

I think the answer in both of these cases
is that it arises from iteration, that the act

of iterating, itself a very simple process
and infinitely repetitious, the act of iterating

a simple equation gives rise to properties
or features that weren't present in the equation

in the first place.

We've seen randomness and unpredictability
arise, and we've seen a really rich complexity

arise from a very simple equation that gets iterated.

This in turn calls attention to one of the
things that I think is different and maybe

noteworthy about the study of dynamical systems.

In physics, and probably elsewhere in the
physical sciences, sometimes when one has

the equation in hand, the story is sort of
over, at least as far as the science is concerned.

That we try to understand a process, and once
we have an equation for it, the process is

understood, story over.

One can then use that equation to make 
predictions build bridges, to build computers.

But the understanding is encoded in the equation itself.

I think the same isn't true for the iterated
logistic equation.

If you have the equation, it's just the beginning.

The equation itself, just looking at the equation,
doesn't make evident that there's a butterfly effect

lurking, or that there's this infinite complexity
and structure in the bifurcation diagram.

Those are features of the system that only
emerge via the act of iteration.

And, very often, the only way to see what the 
system will do is to see what the system will do.

And what I mean by that is one has to just
start iterating and see what happens.

There's not always an analytic method, a 
pencil and paper, a deductive way to figure

out if an iterated function will be chaotic or not.

But if you iterate it on a computer, you can
see very quickly what's going on.

So this is a feature of dynamical systems
that I think is somewhat different than how

equations are used elsewhere. The equation
maybe isn't the end of the story, but it's

the beginning. One has to iterate it and take
a bit of a more experimental approach, to see

what properties emerge from the act of iteration.

And often the properties that emerge are properties 
that weren't present in the original equation.

So, in any event, this brings us to the end of Unit 5.
We've looked at the bifurcation diagram for the

logistic equation.

In the next unit, we'll focus in on this feature
of repeated period doublings, those pitchforks

of sideways U's that come up again and again
and again in the bifurcation diagram.

And amazingly, we'll see that this isn't just
some mathematical or geometric curiosity,

but it actually has some really important
statements to make about actual physical

properties of systems in the material world.

See you then.