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.