In this video, I'll show you how to use

a web program, to make bifurcation

diagrams for the logistic equation.

But before we look at that program,

there are a few things about final state

diagrams that I want to highlight.

So first, let's go back to the parameter

r = 2.9

and I drew the final state diagram, like so.

There's a single attracting fixed point.

It's around 0.65, so I draw it there.

So, as a practical matter we need to tell

the computer two things.

We're interested in the final state, so

we want it to not plot the first 20, or 40

or even 100 or 200 values in the orbit

since we're interested in the long term behavior.

We don't care about the short run, so

we need to tell it how many iterates

to ignore, and then once we think we're

in the long run, when we're at time

100, or 200, or 300, then we need to tell

the program how many points to plot.

Here that seems almost silly, there's just

one point, so we will only need to plot one.

But in general, we don't know what

the final state is, and so we'll need to

tell the computer to plot a bunch of points

so we understand what the behavior is.

For example, if, instead of 2.9, we had an

aperiodic value, here again I would tell

the computer to iterate 400 times, say,

and then maybe plot the next 100.

If I plotted just the next 3 or 4,

I wouldn't see that the orbit was really

periodic. So the two things that we need

to tell the computer: how many iterates

to ignore, i.e. iterate, but don't plot,

and then, how many iterates to plot, once

you've ignored the ones you're going to

ignore.

And then lastly, we'll need to repeat this

process for many, many different r values,

and then put them all together to make a

bifurcation diagram, like this.

And so, we'll need to tell the program

what r value to start at, what r value to

end at, and how many intermediate r values

to do the numerical experiment to make the

final state diagram that then gets

smooshed together

to become the bifurcation diagram.

So, let's take a look at the program and

see how it works.

So here's the program that will plot

bifurcation diagams for you.

You can find the link to this program

right underneath this video on the

Complexity Explorer site, and at the end

of this unit, I've made a separate sub-unit,

a separate page, that has a link to this

program, and the other two programs

from Unit 3.

And there's also a zip file that you can

download, so you can run these programs on

your own computer and they'll work even if

you're not connected to the internet.

So let me show you some of the basic

features of this program,

and in subsequent videos,

we'll explore more deeply.

So, here it is, the bifurcation diagram

for the logistic map,

and I'd suggest you might need to resize

your window, the overall browser window,

so it fits well on whatever computer

you're working on.

One thing you can do is you can change

the size of the plot, how big it is

on your screen. And in order to make it fit

a little better, I'm going to make it

a little bit smaller. I'm going to change

the width to 500, and the height to 250,

and I'll click on 'Update Plot'.

And now, it fits a little better

in the screen.

I guess I can make it a little bit bigger.

Let's do this....

Ok, so here's the bifurcation diagram,

it goes from zero to four.

So, one thing to note is that if you put

the cursor in the plot, it will show you

the co-ordinates, where your cursor is.

So for example, right now the cursor

is at an r value of 2.91,

and an x value of 0.65

and notice that those values change

as I move around.

So if you want to know the co-ordinates,

the r value and the x value for

a particular point, you can do that

just by moving the mouse over there.

Alright, so in order to make this

bifurcation diagram, the program starts by

iterating the seed 0.5, and the seed

you put in doesn't really matter, but

you can change it if you want, to see if it

looks different, and then it will iterate,

how it is now, for 100.

So we'll skip plotting the first 100, and

then it will plot 200, and those are

numbers that we can change, and we'll do

that in later videos.

Alright, so let's see how we can use this

to zoom in.

So you could zoom in by changing the min

and max x and r's, but it's more fun and

much quicker to just highlight a region

with a mouse.

So here I've highlighted this box in blue,

and then if I do 'Update Plot', it shows me

a zoomed-in version of what was in that box

and you can go back, I'm going to click on

the back button, and it will show you the

box that you just entered, and then

forward zooms in there. And I can zoom in

again, let's say I want to zoom in here,

and then I 'Update Plot'.

And I can move back, and forwards through

the plot. Let's zoom in one more time, let

me zoom in on this region.

'Update Plot', there it is, and we can see

bifurcations, so this is some sort of a

period down in the bifurcation.

This splits into two, each of those split

into two, and so on.

So we see the same general shape appearing

again and again and again.

And once you have a bunch of rectangles

selected, you can move through them

backwards and forwards.

So let me zoom in here. This is the

bifurcation diagram now from about 3 up

until 4, and let me zoom in on a different

rectangle. I'm going to zoom in here.

So I'll do that, 'Update Plot', and notice

what's going on in here.

So we have this more or less solid dark

region. That indicates that the orbits are

aperiodic. They're filling up an entire

interval.

But then here we have what's called

a periodic window.

So, where my cursor is now, is period 5.

1, 2, 3, 4, 5.

Let's figure what the r value is for that.

I'll move the cursor here, and I see,

looking at the lower right, that the r value

is about 3.739 / 3.74, and that was one of

the r values that you explored in one of

the quizzes, earlier in this unit.

So this is a period 5 window.

Let's zoom in here just for fun.

'Update Plot'. We can see that same period

doubling structure again.

Let me go back,

and let me zoom in here.

This is another periodic window.

This one is period 123.

1, 2, 3.

The r value in this region is about 3.84,

and that's also an r value that you've

explored previously.

We'll go back again.

Here is the logistic equation from about

2.9 to 4. Let me zoom in on this large

region here.

And so we can see that the regions of chaos,

that's when we have a ... if you were to go

straight up, it would be a more or less

solid collection of dots, indicating an

aperiodic orbit. It's filling up an

interval.

But in these regions of chaos, we see lots

of windows of periodicity. Again, these

are known as periodic windows.

Let's look at just one more.

Let's see what this window is.

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

So here's a region of period 7,

the small little region at about 3.702.

And let me go back.

So this program will let you make

bifurcation diagrams for the logistic

equation, and you can zoom in and out,

and one of the things to notice, and we'll

talk lots more about this in this unit and

the next, is that we see this period

doubling pattern appearing again and

again, and also in regions of chaos, we see

lots of very small periodic windows.

So in the next unit I'll talk a little bit

more about some of the features of this

program and we'll dive, literally, deeper

into the bifurcation diagram.