In this subunit, we'll see that
universality is not just a mathematical

phenomenon but a physical phenomenon.

Chaos occurs in real, physical systems and
so does the period doubling route to chaos.

We can calculate delta for these physical
systems, just like we did using the

computer and the bifurcation diagram.

And we'll see that systems have deltas
very similar, consistent with that number

4.669201. The experiments are somewhat
involved so I'll describe a somewhat

simpler version first to give you the idea
of how one might encounter period doubling

and bifurcations and so on in physical
systems. To do so, we're actually going

to take a short field trip.

I'm going to unhook this camera. 
And move it around this way.

I'm going to walk down the hall, through
here. My office is in a strange, old

building. I'm going to head to the bath-
room. There's something here to take a

look at. Here we go. Here's the water
cooler. Restroom. And here we are.

So what I want to show you is this.
This is a dripping faucet.

I've put a plastic cup under here. I had
to balance it with a sponge so maybe we

can hear the drip. I'm going to take the
microphone and put it closer here and see.

[dripping noises]

So it turns out that dripping faucets can
be periodic and they can be chaotic

So a periodic dripping, faucet, you've
probably heard those before.

They drip in a regular fashion. So they
would drip in a regular fashion, drip,

drip, drip, drip. So that would be a
periodic dripping faucet.

Sometimes you'll get a dripping faucet
and it actually looks like this one is

doing it, and it wasn't before, that drips
in an irregular fashion.

There's no clear or obvious pattern. There
is a different time interval between each

drip. Those are the dripping faucets that
most people find really annoying because

it's harder to block them out from your
mind. If it's a periodic pattern you can

forget about it and sort of subtract that
sound off. It's a little bit like some

sort of noise or some regularity
that you can learn to ignore.

But a dripping faucet that's irregular,
and this one seems to be, has the property

that it's harder to ignore.

You're always waiting. "When is that next
drop going to fall?"

And it's a different time you have
to wait every time.

Ok so let's go back and talk a little bit
more about dripping faucets.

So I'm back in the safety of my office and
let's talk a little bit more about the

dripping faucet. The logistic equation has
a parameter in it, r, that we interpreted

as a growth rate. And we see bifurcations
as we change that growth rate r.

Same story for the cubic equation and the
sine equation. There's a parameter r.

We change it and we see bifurcations.

What's the parameter value for the
dripping faucet?

Well it turns out that the parameter the
experimentalists used is just the flow

rate. So one controls not the growth rate
of rabbits on some make believe island,

but the flow rate of water flowing into
the faucet. So this we could call, r.

It's still a type of rate as it turns out.
And we can imagine varying r, so we have

some dial that we can change r with by
very, very small amounts.

And so maybe initially, for some small
flow rate, the faucet drips in a periodic

fashion with period 1. And that's sort of
the standard drip. It would just go drip,

drip, drip, drip, drip. Same time interval
between all the drops.

Now imagine increasing the flow rate a
little bit, a little bit more, and a

little bit more and eventually one might
observe a bifurcation.

Rather than being period 1, the drops are
still regular but with period 2. And this

might sound something like this: drip drip
[pause] drip drip [pause] drip drip.

So one needs to hear two drips before the
cycle repeats again.

There's no longer the same time interval
between all drops. But it's still regular.

Once you hear those two drops, when you
hear the third one you're back where you

started from. So that would be period 2.

Then, one can increase the parameter,
that's the flow rate, turn that knob just

a little bit more and eventually one will
see period 4 behavior.

And that might sound something like this:
drip drip [pause] drip[pause] drip[pause]

drip drip [pause] drip[pause] drip[pause]
drip drip [pause] drip[pause] drip[pause].

So it's a cycle but you need to hear four
drops before the cycle repeats.

So it's perfectly regular, it's
predictable and so on.

But it now has period 4. And it might even
be possible to turn that knob up a little

bit more, increase the flow rate just a
little bit more and observe a period 8

cycle. So a period 8 behavior in the
pattern of the drips.

It gets harder and harder to see these
different periodic behaviors because, as

we've seen in the bifurcation diagrams,
they occur over smaller and smaller

parameter values, ranges of parameter
values. Eventually when one turns the flow

rate up high enough, one could see chaotic
behavior. So there, there'd be a different

interval between all the drops. There
would be no cycle, it wouldn't cycle back

upon itself. It would be irregular.

So we can analyze those parameter values,
the parameter values at which the

bifurcations occur and calculate deltas
like we did for the bifurcation diagram.

We'll let r1 be the flow rate at which the
drops change from period 1 to period 2.

That's where we see a bifurcation
from 1 to 2.

r2 will be the flow rate at which drops
change from period 2 to period 4.

And r3 will be the flow rate at which
drops change from period 4 to period 8.

So we can calculate deltas for these just
like we did for the logistic equation.

I'll do this in equations in a second.

So Δ1, that's the region of parameter, in
this case flow rate, it's the range of

flow rates in which we have period 2
behavior. Period 2 behavior starts here

and it ends here. So the difference
between these two, that's what we call

Δ1 is the region, all those flow rates
where we have period 2 behavior.

Similar story here. Here we change from 2
to 4. Here we change from 4 to 8.

So it's between these two that we have
period 2 behavior.

And that's what we called Δ2.

So we can then form Δ's just like we did
when we were playing with numbers

on the computer.

So δ1 was defined like that. It's the
range of parameter values for which the

drops were period 2 divided by the range
of parameter values for which the drops

were period 4.

That's given by this. And if one were to
do the experiment, one would see something

that's close to 4.669.

Now you wouldn't expect to see
this exactly for two reasons.

First, there will be experimental error.
It's really hard to pin down the r value

at which these bifurcations occur in the
dripping faucet.

So that's one reason we wouldn't expect to
get exactly that number.

The other reason is that this result is
really only true when we have a large N,

when we're deep into the period doubling
sequence already. We can get there,

experimentally it's really hard to do but
nevertheless as we've seen with the

bifurcation diagram for the logistic
equation and the sine equation, even δ1,

not a large N at all, is very close to
4.7. One wouldn't expect to see 4.669

exactly but one would see numbers that are
very close to this.

So this is a way we can see a feature of
period doubling. This universal quality,

this value δ=4.669201. We can actually go
out, do a real experiment in the physical

world, not a computer experiment,
but a real experiment with real faucets

and real water and come up
with a number like this.