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.