Here's the program on the Complexity Explorer site that will make time series
plots for the Hénon map. This is the same program we used in Unit 7.
Links to this program and all the other programs from this unit are collected in
a subunit called "Links to Programs". It's there on the right hand navigation bar.
We'll be using a bunch of different programs in this unit.
Time series plots for the Hénon map, we were looking at a=0.8.
I'll use 0.8 for a, 0.3 for b, and let's make the time series plot. There it is.
That's what I showed you a screen capture of previously.
Going to look at the phase space, or the x-y plot. We're seeing a lot of transient
behavior as it bounces around before it gets to the final value. If we want to
think of this like a final state diagram, just looking at the long-term behavior.
Let's go back up here. I'll have it plot 4,000 iterates and then have it skip
all but the last 50. So 3,950.
So it'll plot in the time series plot 4,000 but on the x-y plot it'll only
do the last 50 because it'll skip the first 3,950. Let's do that.
It takes a little while to plot all those points and we'll get this big purple blob,
there it is. Let's go down here and we can see, indeed, the long-term behavior is
just those two fixed points, excuse me those two periodic points.
So here's the one that is about (-0.38, +0.38) and that's right over here.
We drew that in pen in the previous video.
Here is the other periodic point (1.26, -1.2).
Notice that the zero axis is right here, and then here.
So since they diagram period 2, we see two dots, just like we did before with the
Logistic Equation. It's just that now the dots live in a plane instead of on a line.
We'll now look at the Hénon map for different parameter value. I'll let a=1.4,
which is the default on this program, these are the parameter values that Hénon
considered when he first did this work. We've done this before but we'll do it
again. Let's make the time series plot. And we see some apparently aperiodic
behavior. It certainly looks chaotic.
We're used to seeing shapes like this from our study of the Logistic Equation.
So then we ask, "what do these points look like if we plot them in the x-y plane?"
For a periodic value we saw that we just got two points.
And in that case, the two dimensional picture was the same
as the one dimensional picture.
So periodic with period two there'll be two points in the final state diagram,
that's the long-term behavior.
It doesn't really matter if those two points are on a line, like they were for
the Logistic Map, or on a plane like they are for the Hénon Map.
So what if we plot the x and y values against each other for this.
For the Logistic equation, when we did that we saw that they filled up the line
interval, the dots kept getting denser and denser and denser on that line.
So we might expect to see a similar sort of blob that the points will fill up two
dimensional space; that the grid will eventually get completely covered in with
dots, left and right, up and down.
So let's see if that is indeed the case.
So here are the two time series. And here are the points plotted in the x-y plane.
There's just 40 of them so we can't really see that much what's going on.
The thing to note is that it doesn't appear that they're moving about just at
random here. There's a whole region in here where there are no points at all.
It looks like it's making some sort of a pattern.
So let's ask the computer to plot more points. I guess I'll do what I did before.
4,000 iterations but this time I'm going to skip just the first 50.
So we'll ignore the first 50 and then plot another... Let's make this 2,000
iterations just to be safe. It'll calculate 2,000 iterates and plot all
but the first 50. So the computer is thinking. There it is. Alright, so the
time series plot looks like what we're used to for chaotic time series plots.
It's an aperiodic orbit. It's bouncing all over the place. Big purple blobs.
But let's see what happens down here. And here we're seeing what we suspected
before, is indeed the case. It is not the case that these blue points fill up this
entire grid. Instead they seem to be filling up this sideways U or, to me it
looks like a boomerang, but this weird sort of shape here.
So this shape, the thing we're looking at is a strange attractor.
So let me say a little bit about what a strange attractor is and then I'll
illustrate that graphically in a number of different ways.
So a strange attractor. The first thing about a strange attractor is it's an
attractor. That seems sort of obvious but it's worth remembering.
What that means is, is that nearby points get pulled into this shape.
We've seen cycles that were attractors and that means that all, or almost all initial
conditions get pulled into that attractor. It's the same thing here; an attractor.
Nearby initial conditions will all get pulled into it. But now the
attractor is chaotic, it's not periodic like the attractors we've seen before.
Ok, so that's the second thing about strange attractors, is that the motion on
the attractor is chaotic.
And the other point for the Hénon Map that I want to emphasize is that these points,
as we do iterate 1, iterate 2, iterate 3, they jump around on this.
So this ends up being a smooth curve and I'll show you a better picture in a second
but the points don't move in a smooth way.
We can see that here. The points aren't moving around in any sort of smooth way,
x vs. t, y vs. t.
So if we were to plot this in slow motion, we'd see a dot here, and a dot here
and a dot here. Only gradually would this shape emerge out of the mist.
I have much more to say, or to show you about this attractor but first I want to
verify this claim I made that motion on the attractor is chaotic.
So to do that we'll open up another program and the link to that is also in
the navigation bar on the right: "Links to Programs", it's the second one under
the Hénon Map.
What we'll do is we'll plot two different initial conditions and we'll look for
evidence of the "butterfly effect."
Here's that other program, this will also calculate time series plots for the Hénon
Map. It'll also make plots in x-y space.
But now it'll do so with two different initial conditions and it'll plot them in
two different colors.
This is very similar to the program we used for the Logistic Equation when we're
looking for evidence of sensitive dependence on initial conditions,
the butterfly effect.
So, again, I'll let the parameter a=1.4, and b=0.3, and here are two different
initial conditions, (0.2, 0.2) and (0.21, 0.21).
So very close initial conditions Let's make the time series plot.
Alright, so here is the x-time series. And this general phenomena, I hope,
looks somewhat familiar.
The two itineraries for two initial con- ditions are plotted in green and purple.
They start right on top of each other.
But pretty quickly, after about 10 or 12 steps they move apart and after that
they are behaving completely differently.
Although, notice they kind of
come back in alignment here and then they go out of alignment there.
We saw that general phenomena in the Logistic Equation too.
Similar story in the y vs. t graph.
That we're seeing two nearby initial conditions that start off right on top of
each other but then they get pulled apart. Let me show this a little bit more
dramatically. I'll do what I did before. Now the initial conditions are REALLY
close, 1 part in 1,000 or something. Now the initial conditions are (0.2, 0.2)
and (0.2001, 0.2001). I might have to go out a little farther. Let's do 100
iterations. Make the time series plot. There it is. Didn't need to go out that
far, let's do 50. Here we go. Similar to what we've seen before. I increased the
similarity between the two, sorry, let me say it in another way. I decreased the
difference between the two initial conditions. Now they are very, very close
and indeed we can predict farther if we're viewing green as a prediction and purple
as reality or vice versa. But increasing the resolution, or the accuracy, by a
factor of 1,000 only lets us double or not even quite triple our prediction time.
So again, this is the butterfly effect. Very, very small differences in initial
conditions can make a very large difference in the behavior of the orbits.
And we see a similar story for y. But now let's see how does this play out in the
x-y plot. So there it is. Notice that the green and the purple are both filling up,
they're both starting to trace out the same shape. So let me try to illustrate
this point. We'll go back here. Again, I'll do 2,000 and I'll skip 50.
Actually let me just do a 1,000 and I'll skip 50. So the computer will think for
a moment. There are the two time series. Purple and green right on top of each
other. Familiar looking mess. And there is the Hénon attractor. So we see that the
initial conditions, they start about the same but we know the butterfly effect
pushes them apart very quickly. So we're seeing two different initial conditions
ending up in the same shape. It looks like there is more green than purple because
green is plotted second. So there are a lot of purple points underneath green
points. So again, what we have here is chaotic behavior, aperiodic. It's a
deterministic dynamical system, orbits are bounded, they're aperiodic and they have
sensitive dependence on initial conditions. But we also have this attractor structure.
Attractors indicate a type of stability. Different initial conditions end up at the
same attractor. But here the attractor is this funny bent shape in two dimensional
space and the motion on the attractor is chaotic whereas previously the motion on
attractors, like a period 4 attractor, has not been chaotic it's been periodic.
Let me do one more thing to illustrate in just a slightly different way this
phenomenon of an attractor that itself is chaotic. Back to the program that plots
the time series and x-y plot for just one initial condition. I'm going to ask it to
do 500 iterations and I'll skip the first 50. Here we go, I'll make the plots.
There they are, x and y both aperiodic but plotted together they appear quite
ordered. It's this sideways U or boomerang shape again. So if I change the initial
condition. Let's say I change initial x from 0.2 to 0.7. If I do that, that's
going to make this x plot look quite different because we have a different
initial condition. So let's see that. Watch this plot down here change.
But the plot down there looks the same. The points might be in a slightly
different place, like there might be a point here instead of there. But the
general structure of the shape is independent of the initial condition.
That's just another way of saying it's an attractor. Multiple initial conditions end
up on this shape. Once they're on this shape they bounce around chaotically.
Let's do this maybe just for one more. I don't know. Let's get a better view of
the attractor. I'll plot 1,000, initial condition of 0.6. There's the time series.
But there is that shape again. And if I change this to 0.66. The time series right
below will be different. Sensitive dependence on initial conditions.
There it is. It's just changed. It'll give me a different time series. But the
attractor structure stays the same. The exact points you might visit on this shape
will be a little bit different because of the butterfly effect, because of sensitive
dependence on initial conditions. But the set of points that are available to bounce
around on this attractor stay the same. So again, this is a strange attractor. It
is a stable structure in that, it's an attractor, lots of orbits get pulled to it
but once one is on the attractor, the motion is chaotic, aperiodic, and it has
sensitive dependence on initial conditions.
So what I want to do next is go to a different program that will let us plot
many, many points and let us zoom in on the structure of the attractor and we'll
see what is deep inside this shape.