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.