So we've been looking at period-doubling

and here again in this picture

we have period-doubling

from two to four, four to eight, and so on.

Capital delta is the range of parameter values

for which we have period 2.

Delta two is the next range,

delta three is the next range,

and we define these ratios

this divided by this,

this divided by that,

to be these little deltas.

And for the logistic equation,

we found that delta one is about 4.75,

and delta two is about 4.65/4.66.

In the quiz that you just did,

you did the same thing for the cubic equation.

And you found 4.419 and 4.618.

So the numbers for delta one and delta two

for these equations aren't similar--

sorry, ARE similar, but not identical.

However, if we kept going on and calculate delta three

and delta four and delta five and so on

these numbers would get closer and closer to each other.

So let me write that,

we're interested in delta n--

which is capital delta n over capital delta n plus one.

And this is going to go to the number 4.669201.

So for both of these equations,

as we let n get larger and larger,

we get closer and closer to this transition point,

these ratios approach 4.669201.

So in the large n limit, this number is just known as delta.

And we would say that delta is universal.

so delta is universal,

and what that means is

that it has the same value for

a very large family or class of functions.

Let me state this a little bit more carefully,

since it's a crucial result.

So again, the phenomenon of universality:

We have delta n approaching a number delta,

and we say that delta is universal.

And the result is

that this number delta is the same,

4.669201 for all iterated functions

that map an interval to itself

and have a single quadratic maximum.

So let me say a little bit about these conditions.

So if the function maps an interval to itself,

that just excludes the possibility that orbits go off to infinity,

either positive or negative infinity.

So the logistic equation is an example.

It mapped the unit interval to itself,

meaning numbers between 0 and 1

remain between 0 and 1.

Let me draw some pictures to illustrate this:

What does it mean

for a function to have a single quadratic maximum?

So single quadratic maximum...

So our favorite example, the logistic equation,

that's just an upside down parabola.

It has a single maximum right here

and it's quadratic because well,

it's a parabola, its a quadratic function.

Here's another example though.

This is let's see...

the cubic function looks something like this.

It's not symmetric

and the function is not exactly a parabola.

However, if we look closely near the maximum,

it would become more and more parabola-like.

So it's locally parabolic.

And in calculus terms, this just means

that the second derivative doesn't disappear.

So most functions that have a peak like this,

will behave this way.

Let me give some counterexamples.

This function does not have a single quadratic maximum

because it does not have a single maximum--

it has two: a maximum here and then there.

So this function does not meet the criteria

in that statement I gave on the previous piece of paper.

A few more counterexamples:

A function like this,

this has a single maximum, a nice peak,

but it's not a quadratic maximum.

If you zoom in on this point,

it doesn't start to look like a parabola,

it keeps looking like a point.

The idea is that this is infinitely-sharp.

Here if you zoom in on this,

you can make it look as close to a parabola as you wish.

So this does not have a single quadratic maximum

because, well, it has a maximum,

but it's not quadratic--it's pointy!

Here's one more example.

It has a maximum, but it's flat,

so it doesn't really have a single maximum value.

And if you zoom in on this line,

again, it won't look like a parabola,

it will look like a line.

So this is also not a function

with a single quadratic maximum.

So any function with a single quadratic maximum

it could be a parabola, a cubic function,

sine functions, various exponentials,

there are many many examples of such functions.

And this is a pretty generic criteria

in that if you just draw a function by hand

or cook something up that has a single maximum,

odds are it'll be smooth

and this criteria will hold.

So this isn't a very restrictive criteria.

And there's a vast number of functions that meet this criteria.

So let me state this once more

This property delta,

which is a feature of how the

sideways used those shapes on the bifurcation diagram,

how they're related to each other,

how much smaller each one gets

as we get closer to this transition,

that this geometric quantity delta is universal.

It has the same value, 4.669201,

and it goes on and on for all iterated functions f(x)

that map an interval to itself and

have a single quadratic maximum.

So again, this is not a very restrictive criteria.

So any function you come up with

that meets this very mild criteria,

you can make a bifurcation diagram

find the bifurcation points like we did,

calculate the deltas,

go deeper and deeper

and let n get larger and larger,

and this number will appear.

So this number is a property of all of those functions.

So I want to emphasize

just how amazing this result is.

We started with the logistic equation,

just about as simple a nonlinear equation

as one could imagine,

and we saw that we had this bifurcation diagram

with an amazing amount of complexity,

but also with some regularity--

we saw those pitchforks,

those sideways Us,

repeating again and again and again,

and we noticed that there's

sort of a geometric similarity to them,

that the next pitchfork in the sequence

is smaller than the previous one,

but maybe smaller by the same factor.

So that led us to investigate that idea quantitatively.

And we defined these lowercase deltas

as that ratio of one pitchfork length

to the next pitchfork length.

And we found that that ratio approaches a constant number,

this number 4.669.

And that's an interesting result for the logistic equation.

It's a practical mathematical oddities tells us about

the geometry of this particular bifurcation diagram.

Then we looked at bifurcation diagrams

for other equations--quite different equations:

sines and cubes as well as parabolas--

and we saw the same general features of the bifurcation diagram,

but then if we calculate these deltas,

and we kept calculating them

to larger and larger periods,

we would find again the same number appearing--4.669.

So that's where things started to get really weird--

that we're seeing the same number appearing

in very different equations.

So that tells us,

at some level the equations,

or the details of the equations don't matter.

There's some broader or overarching

feature of these systems thats

independent of the particular equation

that we use.

So that's something deep and

something pretty surprising I think.

Let me mention just a tiny bit about

the history of this idea and this realization.

The results I presented date from 1978.

They're usually attributed to Mitchell Feigenbaum,

and American physicist who discovered

this property and then did some

analytical work to try to understand it further.

But it was also discovered independently by

Charles Tresser and Pierre Coule,

around the same time,

also published in 1978.

So this is a relatively new result.

Another thing I want to mention is that

I presented these results experimentally--

the result of doing some numerical work on a computer,

but there's a lot of very elegant

and very powerful analytic work

that calculates these numbers--4.669201--

and explains why this number appears again and again.

The mathematical framework for

carrying out that analysis is know as

a renormalization group,

or just renormalization,

and I'll say a little bit about that

in a subsequent unit.

I can't explain it in detail.

It's just too much math for the level of this course.

But I can maybe give some sort of an argument for that.

But before I do that, I want to mention the next thing,

which is,

that this result is not just mathematics,

So it's an amazing mathematical result,

these one-dimensional functions

have these beautiful bifurcation diagrams

with the same branching,

or same sort of fork-ratio in them.

But this is physics as well.

Period-doubling occurs in real physical systems,

and one can go out and measure the rates

at which those period-doublings occur.

And one finds again, this number 4.669.

So I'll describe that result

in the next lecture.