So, I think the phenomenon of universality
in physical systems is incredibly amazing.
We see period doubling in physical systems
and that period doubling is described by a
delta, and that delta has a value that is
the same as for these one-dimensional
systems that we started studying with -
the logistic equation, and so on.
If you're not already convinced that this
is amazing, let me try to convince you by
going over the line of reasoning that led
us to this point. So, I started a long
time ago in this course talking about the
logistic equation. And when I presented it
I said don't take this equation very seriously
- it is certainly not a realistic model of
how rabbits might grow, or how almost anything
might grow. But it's more like a characature
or sketch or cartoon - it gives us some
rough picture of how a population might grow
where there is some limit to the growth.
And then we found that there is surprising
richness to this equation, amazing richness
when we form the bifurcation diagram and
zoomed in again and again and again and saw
these fractal patterns and complicated behavior.
But we might have thought...it would be very
reasonable to think "well we're just playing
with a mathematical game - it's just an
equation that somebody told a story about
rabbits to motivate - but it's just a little
equation and it's really neat that it does
all this stuff - but what might it say about
the real world?" And we talked about that sum,
and I appealed to the argument that this
shows that even the simplest possible...
or just about the simplest possible equation
...iterated function...can show the butterfly effect
and sensitive dependence and that's an
important realization and that certainly is.
But still the logistic equation seemed just
like a little toy model - a fun thing to play with.
Then, in this chapter, we started looking at
...or in this unit...we started looking at
bifurcation diagrams for different equations
- the sine equation, the cubic equation, we
could have done lots of others - and we saw
that the bifurcation diagram was kind of
similar. Then, I presented the notion of this
quantity delta, looking at the ratios of the lengths
of period two and then period four and then
period eight regions. And we saw that
the same number appeared - 4.669201 - so
that all of these different functions turn out
to behave the same...not exactly the same,
but they undergo this transition to chaos
from periodic to chaotic behavior in the same way.
We're still in the realm of math. It's an amazing
mathematical fact that we're still in the
realm of math. We might wonder what this has
to say about the real world. So now,
experimentalists, scientists, and physicists have
gone out and actually measured period-doubling
in a whole bunch of different physical phenomena
and they've calculated this quantity delta -
this ratio of period one to period two,
period two to period four, and so on. And they
find numbers consistent with that predicted by
this theory - 4.669. So, we started with a
ridiculously simple, deliberately unrealistic
equation, and nevertheless, that equation
contains some information in the form of a
number that one can go out and measure in real
physical systems. So this surely tells us something
important and profound - that nature and mathematics
are somehow constrained in the ways that a transition
from periodic to aperiodic behavior - from
periodic-ordered behavior to chaotic
behavior - can occur. That when it undergoes
period-doubling, almost always what it's going
to do in the large-end limit, it's going to do this
transition in the same way. So all of these different
phenomena - period-doubling in rabbit equations
on the computer and in sine equations on the computer
, and in electronic circuits, and in convection rolls,
and more - all have some similar feature about them.
There is some way that this transition is constrained
and that certain featyres can only occur in one way. So this also
says that some of the details of our mathematical models
apparently don't matter because we see the
same behavior across a very wide range of systems.
That's what it means to say that this quantity
delta is universal. So, of course this leads to
a bunch of questions - how can this possibly be?
And there are at least two parts of that question
I think. The first is a mathematical one -
how can it be that all these different one-dimensional
functions have the same delta? That they undergo
period-doubling in essentially the same way -
this 4.669 number. The other question is
what in the world would a dripping faucet have
to do with a one-dimensional equation? Or fluid
mechanics where we have convection rolls wiggling
around in a little box? These are surely high-dimensional
systems - you would model them with more than just
a one-dimensional equation, and they're also
continuous systems - they're not discrete like
iterated functions. The second question - what
one-dimensional iterated functions might have to do
with higher-dimensional real physical systems - I'll
discuss in the next several units. In the next sub-unit ,
the next set of lectures here, I'll try to give some feel
for how it can be mathematically that there are
these similarities among all of these different
one-dimensional functions.