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.