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.