In this video we will explore

the bifurcation diagram.

And see some of the complexities

and patterns that it holds.

So here again is the website

that plots the bifurcation diagram for us.

And I have resized it,

so it more or less fits inside the screen.

So, the first thing to call attention to

as we have already seen is that

this period doubling repeats again and again

So period doubling, so here we have

period 1 behavior.

And that goes over to period 2

as R is increased.

Then, period four.

Then period eight.

and we see that period doubling again

and again.

In...sorry I'm having a hard time

with the mouse here.

We see that period doubling again and again.

As we zoom in.

So, here I have zoomed in a bunch.

Let's zoom in

I don't know this region here.

And there is a period doubling again

I'm going to zoom in here.

And now I'm starting to lose resolution.

I'm sort of missing some points.

So I'm going to plot two thousand

instead of two hundred.

And I'll zoom in here.

Opps, let me try that again.

See, what's---what's going on in that square.

And, we are seeing

this feathering or smearing.

So I should probably skip more points.

Because, it's taking longer to reach the long

term behavior.

So I'll try skipping a thousand points before

the plotting begins.

And that clears that up.

And maybe, I'll try plotting 10,000 points.

That will take a little while hopefully

not too long.

That darkens it up.

And let's--let's zoom in a little bit more.

Let's see what's going on up here.

Try that again. Alright let's zoom in there.

Period doubling again.

Let's zoom in there.

Again starting to see some smearing out.

So I'm going to skip 10,000

before I start plotting.

And that crisps things up again.

I'll try doing one more zoom.

There we go.

I'll zoom in.

And I need a few more points

So I'm going to plot 20,000 points.

Maybe I'll do a little more.

I'll do 40,000.

make it a little bit darker.

Then I'm going to skip plotting--

--I don't know 20,000.

And that should I don't know.

A little bit clear up

that smuge. Maybe I'll skip 40,000 now.

So for each final state diagram,

it iterates for 40,000 then plots

the next 40,000.

And here is the pattern that we see.

So, this looks alot like

very, very similar to

a larger portion of the bifurcation diagram.

So this has a sort of fractal structure to it.

And that we see small copies of the whole.

The large object.

And we have zoomed in

a tremendous amount.

The difference from the range of R values.

that we are looking at

3.7432448284 to 3.7432448352

So it's only in the last three digits

that there is a difference here.

So let's think actually for a moment

about just, just what a small range this is.

Let's see if we can come up with a way

to visualize this.

So on the bifurcation diagram,

that I just showed on the screen

the R value started at this.

and ended at that.

So they are very, very close together.

Let's see what the difference is

between these two.

To do that I'm going to have to subtract

and I'm not very good at subtracting.

So let's see if I can do this.

2-4. I'm going to borrow one.

12-4. Let's see that is 8.

That five turned into a four.

4-8. I need to borrow 1.

Let's see 14-8 is 6.

Then I got 2-2. And lot's of zeros.

So that's a very, very small number.

Let's put it into scientific notation.

That maybe helpful

1,2,3,4,5,6,7,8,9.

So this is 6.8 times 10 to the -9.

A tiny, tiny range.

So if we imagine we are measuring R.

We usually only think of that as a growth rate.

To make this more concrete.

If we think of this in meters.

This would be about 6.8 or 7 nanometers.

And that's the range we are looking at

this bifurcation diagram.

Over a range of about 7 nano meters.

And that is tiny,

maybe the size of a large molecule,

several large molecules.

We have zoomed in a tremendous amount

is the point I'm trying to make.

And here is another way to think about this

how much we are zoomed in

so, in the current zoom in.

On the screen we were looking at

before we came over to this piece of paper,

we had a R range of this

6.8 times 10 to the -9.

And that took ten centimeters.

So, my laptop screen where I was

cheating that maybe the image was

10 centimeters across.

And on your video, depending

on how you are watching the video,

might be a little bit smaller.

If you are screen it might not be

exactly this size but the point is

we are trying

to get an order of magnitude estimate.

It is closer to 10 cm than 1 or 100.

So, this was the scale we are in now.

And suppose we wanted to view

the entire bifurcation diagram at this scale.

So right, so this 6.8 was an R range,

If we wanted the R range to be four.

How many centimeters would that be?

So it is going to be very, very large.

We have zoomed in, zoomed in so much

that the bifurcation diagram is very, very large.

It has zoomed off the screen

many, many times.

So let's see how large it is.

So let's solve for this question mark.

So I can cross multiply and divide

and I would get the following.

The size of....

So at this scale the size

of the original bifurcation digram

would be 5.9 times 10 to the 9 cm.

That's almost six billion centimeters.

Which is about 5900 km.

That would be a huge distance.

To put that into perspective,

the circumfrence of the earth is only

40,000 km.

So, to recap what this means is

we started with a bifurcation diagram

which went from 0 to 4.

And, we zoomed in again and again

and again.

And we kept seeing this pitch fork structure,

this branches that split into branches

that split into branches

We saw that at different places

in the bifurcation diagram.

And we kept zooming in and zooming in

until we saw one of the those pitchforks,

one of those splittings

over this tiny tiny range of R values.

And at that point, we zoomed in enough

that the original diagram

that it would have gotten so large

that it would have stretched around

the earth's circumfrence.

And we are zooming in and almost

unimaginable amount

into this mathematical objects

this bifurcation diagram.

And we are seeing this structure repeat

again and again

So, here is the bifurcation diagram again

And if this bifurcation diagram

is about ten centimeters.

This small, tiny, tiny portion

of the bifucation diagram

looks to be about ten centimeters.

Then at this magnification

the full bifurcation diagram

from R=0 to R= 4.

Would stretch to around the circumfrence

of the earth.

Alright let's zoom back out.

We have zoomed in a great deal.

So I'm hitting the back button again

and again.

Up we go.

And there is the orginal

bifurcation diagram.

In fact it has made it a little larger

Now let's go forward

Let's zoom back in

to see that one more time.

So zooming in, zooming in, zooming in.

Some of these steps increase resolution.

Or, the number of points I skip.

Zooming in more and more.

The full diagram gets larger and larger

as I do so.

And, let's see there was the last step.

So at this magnifcation,

the full bifurcation diagram

stretches around the earth.