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].
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.