In this lecture we'll examine
the period-doubling route to chaos in the logistic equation in considerable detail.
So here's the bifurcation for the logistic equation
R goes from zero to four
Period one goes to period two to four and so on
and then all this interesting behavior in here.
Here's a closer view of that. Again, you've seen these before.
This is from three to four. Period two to four to eight,
then regions of chaos
periodic windows
more chaos
and so on.
And what I want to do now is focus on just this particular portion of the bifurcation diagram,
where the periods double
and then eventually give us chaos.
So I'm going to zoom in on that even more
and we'll look at this bifurcation diagram.
So now this goes just from 3 to 3.6.
It's period one over here.
At 3.0 it becomes period two.
Then period four, then period eight, and so on,
and by the time we're over here
in these dark regions,
the function is chaotic
and it has sensitive dependence on initial conditions.
So if one looked for the butterfly effect over here,
one would see that yes, at 3.59 or whatever that is,
there is indeed chaos, aperiodicity, and the butterfly effect.
Over here there is not a butterfly effect.
The easiest way actually to test for that
is with a Lyapunov exponent.
Over here the Lyapunov exponent is positive.
And over here it's negative,
meaning that there's not sensitive dependence.
In any event,
there's a transition from periodic nonchaotic behavior
to chaotic behavior,
somewhere right around here.
So we'll look at what happens as we approach
that chaos-order transition point.
So I want to start by looking at the length of this.
So how long--in the sense of,
what range of r values gives me period two behavior,
what range of r values give me period four,
what range of r values give me period eight?
So I want to look at what these r values are
where these transitions occur.
And
I'll call the r value for this first transition r1
and that's 3.0.
So this transition
--I'll put a little arrow there--
occurs right when r is 3.0.
Then there's a transition that occurs here.
This is transition from period two to period four.
And it turns out that this occurs at
r is about 3.44948.
So I figured that out
by zooming in on the bifurcation diagram
looking more and more closely at this point
and trying as accurate a value for
where this line splits in two
as possible.
This was the intermediate,
one of the beginner-intermediate problems
from last unit's homework.
OK, there's also a transition then over here.
And this is where the behavior goes
from period four to period eight.
And I'll call that r3--
It's the third bifurcation.
And that turns out to be
at around 3.544089.
So that's where this transition occurs.
To the left of this transition we have period four.
A little to the right of this number
we have period eight.
And then lastly,
there's a transition from period eight to sixteen.
And that's really hard to see
on this diagram.
We'll have to zoom in a lot.
But if we did,
we would find that
that transition occurs where r is about 3.564407.
So we have four bifurcations,
from period one to two,
two to four,
four to eight,
and eight to sixteen.
And they occur approximately at these r-values.
And I'll call the first r-value r1
and then r2, r3, and r4.
We're interested in this length.
What range of r-values give us
period two behavior?
And I'll call that length
capital delta1.
So let me draw that here.
So I'll do it in the middle.
So this triangle is the capital Greek delta
and its just r2 minus r1.
This length minus that length.
So this is the range of r-values
for which we have period two behavior.
Then I'll...
Can do a similar thing for
period four
and then also
the period eight region.
So this line segment here,
this length right where my finger is,
is delta two.
and this small line segment here,
notice the regions are getting smaller,
is delta three.
And delta two will be r3 minus r2,
delta three is r4 minus r3.
Let me write that.
This is r4 minus r3,
and delta two is r3 minus r2.
OK,
so we've got these deltas.
This length, that length,
and that length.
And we observe,
as we've seen the bifurcation diagram,
these lengths are getting smaller.
So what we'll do next is
we'll calculate the ratio of this length
to that length.
And we'll call that lowercase delta.
So,
delta one is just
over delta two.
So little delta
--that's the lowercase Greek letter delta--
is just this length divided by that length.
And in terms of the R's,
let's see what'll that be?
That will be delta one,
which is r2 minus r1
over r3 minus r2.
So we have numbers for these
and we can plug these numbers in
and we can get a numerical value for delta.
So
let me do that.
So this will be
So I've just plugged in
the numbers I have for r2, r1, and r3.
And there they are.
And now, its subtraction
--I always like doing subtraction
on the calculator--
well let's see this one
I don't need to do.
3.44 blah blah blah
minus 3.0
is just .44 blah blah blah.
So this on top is 0.44
and blah blah blah is 948
And then this subtraction on the bottom
I better do on a calculator.
Let's see here
So I've got 3.544089
minus 3.44948
and that gives me 0.094609.
And the last thing to do
is to carry out that division.
I'll do that quickly.
Let's see
0.44948 divided by0.094609
and I get 4.751.
Okay, so
that's a bunch of calculator work
not tremendously exciting I don't think
but let's just interpret this number 4.751.
What that says is
is that this length is 4.751 times longer
than this length.
That's all it means.
We can do a similar thing now,
comparing this length to that length.
And if I do that.
Delta two is just
capital delta two over delta three.
It's this length
divided by that smaller length.
And I can define that in terms of R.
Carry out all this stuff,
there's no point in redoing all this subtraction.
If I do that,
I get the number 4.6564.
So I've skipped a lot of
intermediate steps here that you can just
do on a calculator reading off these numbers.
And we get this number.
And what this means
is that this length
is 4.65--about 4.66--times larger
than this length.
And we could keep doing this,
we could look at delta three,
delta three over delta four,
delta four, delta four, delta five,
and so on.
So we can ask ourselves
what happens
--we can try to do an experiment--
what happens,
as I go deeper and deeper into this transition,
the periods double more and more,
what happens to this ratio?
So here's a simplified view of what we just did
here's a schematic of the bifurcation diagram
period two to four to eight to sixteen
delta 1 is this length
delta 2 is that length
and so on.
And we're looking at the ratios
How much larger is this than that?
You call that delta 1,
and that turned out to be about 4.751
there's an approximate here because
this was an experimental result
and there's likely some numerical error.
It's very hard to pin down the exact value
of this transition
But this is pretty good.
And then delta 2...
that's how much longer is this length
than that length.
And that was about 4.6564.
And we can keep going
and define delta n as follows:
So it's just the ratio of
one periodic region to that
Let me say that again:
It's the ratio of these two deltas,
this is the length of one periodic region
this is the length of the doubled periodic region.
And what one finds
that as n gets large,
this approaches the number 4.669201.
So as we go deeper and deeper
into this period-doubling,
the lengths of these pitchforks
or branches
approaches a constant ratio
so that each branch--
this branch is about 4.669 times
as large as that one.
This is 4.669 times as large as that one,
and so on.
This doesn't become exact until n becomes large,
but its a pretty good approximation
even for small n,
even for the first set of period doublings.
So this says that as we get
closer and closer to this transition point--
the transition from periodic to chaotic behavior--
there's a regularity to that,
that we see new periods appearing
at this constant ratio.
So now we have a way of characterizing
the period-doubling route to chaos
in the logistic equation and
we can see if
the same thing holds true for
other bifurcation diagrams.
There are two quizzes following this lecture,
where you'll carry out some of this analysis,
but for the cubic equation
instead of the logistic equation.
In order to do that,
you'll need a program that makes
bifurcation diagrams for the cubic equation,
and there's a link to just such a program
in the section titled,
"Bifurcation programs" or
"Links to bifurcation programs"
on the navigation bar on the complexity explorer site.
So find that program and open it up
and give the next two quizzes a try
and see how these deltas work for another equation.