Hello and welcome to Unit 8.

This unit is on strange attractors.

Strange attractors are mathematical 
objects that reside in phase space

that combine elements of 
order and disorder,

predictability and unpredictability.

In some fun and interesting ways.

I'll introduce strange attractors through

a couple of examples. We'll look at the

Hénon map and the Lorenz equations,

which we studied previously on unit 7

and we'll see that they both

posses strange attractors.

And I'll show you what that means in

different ways of looking at them.

Then I'll talk a little bit more generally

about transformations and phase space

and in the summary for this unit I'll give

some remarks on why I think strange
attractors

are interesting and fun and important.

So let's get started by looking again at

the Hénon map.

So here is the Hénon map, it's an iterated
function

a two-dimensional iterated function.

There are two variables, X & Y.

And the next X & Y value is just a

function of the previous X & Y values

and this tells you what the function is.

So you start with an initial condition in
an X & a Y

and then this gives you a time series

of a X's and a time series of Y's.

So we can choose some parameter
values

and to start us off
I'll choose "a" at 0.8

and "b" of 0.3 and we need 
to choose initial

conditions, I think I chose (0.2, 0.2)
for X & Y

and then we just apply this rule 
over and over again

obediently iterating the function as

we've done throughout the course

and If we did that on a program on the web

which I'll show you, again in a second but

just here's a screen-shot from it.

One can see that the orbit pretty quickly

approaches a cycle period of two.

So this is the time series for X.

This is the time series for Y and they're
both, wiggling, repeating every two steps.

If one looked closely at the numbers

one would find that it's oscillating

between these two points so we have

an X of about -0.38, so that is this
value here,

and when X is -0.38 it turns out that
Y is about +0.38.

And then the large X value that goes with

the small Y value, the large X value is

about 1.3 it turns out about 1.26 and then

so large X, small Y.

This value turns out to be about -0.12.

It's a plus. So the points are oscillating

between this and this, back and forth

between this and that. So I can draw

a final state diagram like we did for

the logistic equation but now the final

state diagram is two-dimensional because

we have X and Y, instead of just X.

For periodic behavior on the logistic

equation, the final state diagram was two

points on a line, here the final state
diagram will be two points on a plane.

So let me draw that and I'll make a rough
sketch of what this might look like.

Then we'll see on the computer screen.

So I'll do a really rough sketch of this

is -0.5; +0.5; +1; +0.5; -0.5. Ok

So let's see we've got two points,

the first point is X=-0.38, Y=+0.38

so I'll go to the left -0.4 up about 0.4

and that would be the point on the XY

plane, then the other point +1.26 that's

out about here, -.12 that's probably

around here. So again the scales are not

perfect it isn't a beautiful diagram but

the main point is that the final state

diagram for the Hénon map when we have

periodic behavior is just two points on a

plane. And remember the final state

diagram, I'll show you this again,

we're interested just in the long-term

behavior, so I might not plot in this case

the first ten, and then plot the remaining

ten, and then all those points would be

right on top of each other because
they're cycling on the same point.

So we'll go look at the computer screen
version of this in

just a second and we'll look at this.

This set of parameter values and then

we'll go to the parameter values that we

ended with. We'll make a=1.4 and we'll see

a much more 
interesting final state diagram.