In actuality, I don't think anyone has calculated
delta for the dripping faucet.
The dripping faucet has been studied experimentally
quite a bit, and for sure it shows periodic behavior
of different periods, it shows period doubling
and it shows chaotic behavior as well.
That's well-established. But I don't think anybody
has been able to pin down the bifurcation values,
the flow rates at which those transitions
occur to enough accuracy to get a good
measurement of delta.
The dripping faucet is just too sensitive to
vibrations in the room, and so it makes it
very hard to get really accurate results.
However, there have been lots of experiments
that have been able to calculate delta for
different systems that undergo period doublings.
I think they are not as clear to talk about
as a dripping faucet, which is why I did not
begin by talking about them.
But let me try to describe one of those experiments
and then I'll present some results and we'll see how
it compares with this number 4.669.
So the system I'll talk about is convection rolls.
So imagine we have a small little box and
some fluid is inside it.
It could be water, actual experiments were
done with liquid helium or mercury, but maybe
I'll talk about it as if it's water since
that's more familiar, could be air for
that matter, some fluid.
So we have this box, top and a bottom, and
we heat the bottom, so the bottom starts to
be a little bit warmer than the top.
And so initially heat will just conduct through
the box. But eventually, the bottom part of the
fluid becomes less dense, which is what happens
to fluids when you heat them.
And because it's less dense, that fluid will
start to rise.
And that's a familiar phenomena, that's basically
how we get wind outdoors.
It's due to the phenomena of convection, less
dense fluids, they're lighter and they tend to rise.
Now if we're in a small box with a constrained
geometry, what can happen is is that the fluid
as it rises organizes itself into two rolls.
So the fluid kind of rises up the middle and
falls out the back, so it kind of makes this pattern.
And if the box is longer towards you than
it is this way, and if the geometry is just right
you can get two very clear well-formed rolls
in the box like this.
Of course one can't see the rolls because
the fluids are usually clear but one can
measure what's going on, you can measure
the velocity of the fluid using some clever method.
So we have these convection rolls rolling
like this, rolling like this. And the parameter
in this experiment is the temperature difference.
The difference between the temperature at the
bottom of the box and the temperature at the
top of the box.
So as we heat the bottom more, we are increasing
the temperature difference, that's our parameter.
That's our r value, so to speak.
For the dripping faucet, the r value was the
flow rate into the faucet.
And for the logistic equation, the r value
was the growth rate of these mythical rabbits.
Okay, so back to convection. So we have these
nice convection rolls rolling along, and as the
temperature of the bottom is increased, as
we increase that parameter, the convection
rolls will start to have wiggle in them, and
that wiggle will move back and forth along
that convection roll with a particular frequency,
and that would correspond to our period one behavior.
If we heat the box a little bit more, a second
frequency emerges.
So the motion of the wiggles, up and down
that convection roll have two frequencies in them.
And that corresponds to the period two behavior that
we saw in the dripping faucet or in the logistic equation.
Increase the temperature just a little bit
more and then one will see period four behavior,
there will be four frequencies oscillating
up and down the convection roll.
And sometimes one can increase the temperature
a little more and see a period eight behavior.
So the main idea is that convection rolls and
fluid start to get these small instabilities
which oscillate up and down the roll and
the periodicity of those doubles and then
doubles again and then doubles again. And
we can record the temperature differences
at which those period doublings occur and
then we can calculate delta, like we imagine
for the dripping faucet and like we did for the
bifurcation diagrams we studied earlier.
Here is a sketch of this convection roll scenario
I was describing earlier by waving my hands around.
The sketch may or may not be helpful, it's
not a great sketch, but let's see what we can
get out of it.
So here's a box and the box is filled with
some sort of fluid.
The bottom of the box is hot and the top of
the box is cold.
So there are different temperatures, and
the temperature difference is what we would
measure for the parameter.
Inside the box, the fluid rises in the middle,
and then when it gets up top it cools, and as
it cools it becomes more dense and heavier
and so then it will sink again.
And so we have motion like this, and if the
dimensions of this box are right, we get sort
of two rolls, two cylinders of moving fluid.
This one's moving that way, and this one's
moving that way.
So then one starts to see a little wiggle that
might travel up and then down a convection roll.
And that's the behavior that starts off with
a period of one, and then doubles and doubles
again as more frequencies get added to that
motion.
So this is a system that's been quite thoroughly
studied, and pretty quickly after the discovery of
universality in chaos in 1978.
So let me describe the results of doing some
of these experiments.
So here's a table showing the results of
six experiments.
So these three are for convection rolls.
That's the scenario that I just described.
So you have fluid in a box, you heat it from
below, convection rolls form and then you
watch how the frequency of wiggles on those
convection rolls form, how those frequencies change.
It's an easy thing to say, but it's actually
a very very hard experiment to do, and
it took a great deal of ingenuity and patience
in order to pull it off.
The challenge is both that one needs to
control the temperature difference between
the top and bottom very very very accurately
and precisely, and it's very hard to figure out
how to measure those frequencies because
you can't look inside the box.
So an enormous amount of work goes into
just getting these numbers.
So here are three values for delta.
This is just the same delta that we discussed
for the bifurcation diagram, the logistic equation,
but now the parameter is not growth rate of
rabbits, but temperature difference in a physical
experiment.
So here we see 4.3 plus or minus 0.8, 3.5
plus or minus 1.5, 4.4 plus or minus 0.1.
So they're not exactly nailing the predicted
value of 4.669 but we wouldn't expect it,
and again, for two reasons.
One is experimental error, but the second
is we don't expect to see 4.669 until we're
deep in the bifurcation diagram and experimental
limitations mean we can only see a couple of period
doublings.
There are lots of other physical systems that
exhibit period doublings and then chaos, and
here are just three more of them.
So here are two separate experiments with
an electronic circuit.
So one hooks up a circuit with some sort of
a feedback loop, and one can see period 1
behavior, a steady state and then an oscillation,
and then an oscillation with a second frequency
on it and so on. So again one sees period
doubling here.
These experiments, I think, are a little easier
to do, we get 4.5 plus or minus 0.6, 4.3 plus
or minus 0.1.
Here's another experiment that was done,
this was a different type of circuit device,
a transistor and not a diode.
We don't know the details of these experiments,
the main thing I'm interested in is different
physical systems that undergo the period
doubling route to chaos have very similar
deltas and deltas that are within experimental
error, consistent with the predicted value of 4.669.
This data is from a book by Predrag Cvitanovic
called Universality in Chaos, the full citation is in
the additional reading section for this unit.
It's a volume with a few original papers but
mostly reprinted papers, some of the really
important papers established in universality
and chaos.
So there's a table in that book that summarizes
these six experiments and probably about a half
dozen more and gives the citation to the original
experimental work there.
But the bottom line is, lots of experiments
have been done, are not easy to do, some of
them are really hard, these.
One can calculate a delta as defined before
and one gets values that are by and large
consistent with this 4.669 number predicted
by theory and seen in bifurcation diagrams
from one-dimensional functions.