In the quiz from the previous section we should have found something kinda weird when r equals 4.

The orbits don't settle in to a periodic cycle, they do something different.

They continue to behave irregularly and, in fact, there is no period at all.

As you probably guessed, this is chaos.

You've seen our first example of a chaotic dynamical system.

So, we have lots to talk about in the next several lectures.

I will be able to define what chaos is, and also what sensitive dependence on initial conditions is,

also known as the butterfly effect.

We'll start by looking more closely at the orbits of the logistic equation when r equals 4.

So, let's look at orbits for the logistic equation when r equals 4.0.

So, I'll change this to 4.0 and let's make the time series plot

and here is what we see;

we can see that the orbit bounces around, gets very close to the apocalypse, or annihilation population

here crashes then grows, and then jumps around again,

and it looks another crash was happening here

but the key thing to note is that we are not seeing any periodic behavior

for hints of it there are some regularities,

a pattern repeats this crash, 
a growing pattern repeats a little bit

but it never repeats exactly.

Let's look at this for a larger number of iterates

here you see it again, continues to jump around,

it looks like it almost gets stuck here around 0.75, but not exactly

then, again, we see these big falls when it gets close to the apocalypse value

and we can look down on this table of numbers, and perhaps you can look for numbers,

maybe when you did the quiz you looked for numbers that repeat

but you'll keep seeing new numbers, 
this keeps cycling around

Let's do one more thing, 
let's try plotting a thousand iterates.

This will look like one big purple blob.

There it is.

And, I guess, the main thing again to know is that we are not seeing

any sort of periodic or regular behavior emerge

if, say, we get back to 3.2, or 3.1, that was an example we studied, that was a periodic value,

you look at the time series plot there, we can see some regularity.

It's a little hard to see, but it's bounding back and forth between these two values.

Whereas if I go to 4, it keeps bounding around.

I think this will go up to 10 thousand... It takes a second to think

It's calculating, it's plotting

Still thinking, still thinking

I am really stressing my computer up by doing this, I feel bad

Now it's getting stressed again

OK, there it is. That was a long time to wait for a big purple rectangle.

So, this is 10 thousand iterates of the logistic equation, it keeps on bouncing around

let me jump all the way down to the bottom of the screen

there it is. The number still haven't hit any sort of cycle

and we keep seeing new numbers all the time.

So we say that this behavior, rather than being periodic, is aperiodic:

The orbit does not repeat.

So, let's summarize the behavior of the logistic equation for r equals 4.0

like we did for other r values.

The difference is that, for this r value, we find that the orbit is aperiodic. It does not possess a period.

The orbit does not repeat, it keeps cycling around irregularly.

So, I have a puzzle: How can we represent that, on a final state diagram?

So, here is the unit (--), the population is always between 0 and 1, remember 1 is the apocalipse

or annihilation population, and 0 is zero.

So, what are the final states for this?

Well, if we iterate it for, say, 10.000 times and then watched it for another 10.000,

we will continue to see the population, which you can think of as a dot on this line, bouncing around.

So, another way to think of this is, iterate for 10.000

and then put a next 10.000, open up a camera for a long exposure photograph as the dot bounces around

and we would see that the dots would fill up this line,

just like in the previous image the purple dots filled up the rectangle

so, it would look something like this

So, there will be so many dots here that they would fill out the entire line, so it would look like a solid line

of these dots. So, the final states is not just 2, or 4 dots because it is periodic, the final states would be

this entire line, because it is aperiodic, and orbit wanders from

very close to 0 to almost 1. 
So, there is one more thing I should mention before

I am overcome by fumes from this sharpie.

And that is, a little bit about this claim of mine that the orbit is aperiodic

So, this is a result from mathematics that is proven exactly and rigorously for the logistic equation when r equals 4.0

doing so is beyond the scope of this course and requires a good bit of mathematics

it is a standard part of most junior level courses on chaos, or dynamical systems

So, I'm sorry I can't do this proof here, but it is important to mention so that one understands

that this claim is rigorously established.

It is not just a computer result or an experimental result.

It is something one can prove, or deduce, from first principles.

So, this orbit really is aperiodic. There is an infinite number of numbers between

0 and 1, an uncountably infinite number in fact.

And, if you iterate this forever, you will never see the same number twice.

You'll keep seeing new numbers as the orbit moves around.

I think aperiodic behavior in the logistic equation is a somewhat surprising and interesting result.

We make the orbits by iterating a function. We are doing the same thing over and over again.

It is a very repetitious process

But the result is not repetitious at all: it is an orbit that is aperiodic, it never repeats.

So, it's interesting to me, that this very orderly unchanging process produces behavior that

looks so irregular. Moreover, we'll see that the orbits produced by the logistic equation are

in a sense unpredictable. They have the butterfly effect, or sensitive dependence on initial conditions.

And, in a sense, we can even say that the orbits produced by this simple deterministic function are randomness.

So, let's begin digging into these ideas by looking at how two different orbits for the logistic equation behave.

We'll do that in the next lecture.