So we can iterate the Hénon map very similar to how we iterated the logistic
equation. We just start with a value, in this case it's two values and we just get
the next value by applying the function over and over again.
And we get a time series for x and a time series for y.
It's a straightforward task, soon we'll turn to a computer to do this for us.
So now that we have these numbers, what can we do?
Well, as usual in this course, we'd like to know what's the long term behavior of
the orbit? Are there fixed points? Are they stable? Can we have chaos? What does
it look like?
So we can plot the time series for x just like we did for the logistic equation.
And we can plot the time series for y as well. Those will both be one dimensional
functions, i.e. a dimension for x, a dimension for y.
But another thing we can do, is plot x against y. Similar to how we plotted x
against y or r against f for these two dimensional differential equations.
So for the logistic equation we had a final state diagram that was one
dimensional. Here, we can have a diagram that would be two dimensional.
So I'll just sketch this and then we'll look at it on the computer.
So the first point, x=0.2, y=0.4. I might go over 2, up 4, maybe something sort of
roughly like that. And then the next point x=1.3, so that might be maybe out here.
And y_1 is kind of down here. Because x=1.364, y=0.06 so there's that point.
Then the next point now x is negative, x=-0.6 and y=0.49, that looks like, let's
see that would be probably be around here. And let's see, this is point 0, and that
was point 1, this was point 2 and so on. So I could keep plotting these dots, many,
many of them, and see what happens. See if we get pulled into a fixed point or
maybe some sort of a cycle or something.
If I wanted to know just the long term behavior, I might iterate for 10 or 100
times not plot those but then plot the next 100 or something like that.
So this is very similar to what we did for the final state diagram for the
logistic equation, but now instead of a line we have a plane.
We have a plane because we have two numbers, two coordinates, not just one.
So there's a program on the Complexity Explorer site that will do this for us.
So let's take a look at that program now and understand some of its features.
So here's the program on the Complexity Explorer site that will make time series
plots of the Hénon map for us. You can find this program on the site.
There's a link to it in the right hand navigation bar.
I think it's section 7.6 or 7.7 that says something like Hénon map programs.
Shouldn't be too hard to find.
So here's the program. It comes up in a pop-up window that I've resized slightly.
It should look pretty familiar. It's very similar to the program that we used to
plot the logistic equation, the logistic map.
So, first we can control the number of iterations. I'll choose the default, 40.
Our initial conditions, the ones we've been working with; x_0=0.2 but y_0=0.4.
So I'll change that. And the parameter a, there's two parameters now, a=0.9, b=0.3.
And I'll describe what this box does in a second. So let's make the plot.
So this is a time series plot for the x values; the x values change over time.
Time 0, time 1, time 2 and so on. There's the time series plot.
Here's a time series plot for the y value, it also changes over time.
And we can make time series plots for both x and y separately just like we did for
the logistic equation.
Let me slide down here. It also produces a list of numbers.
Here's our initial condition, (0.2, 0.4). The first iterate was (1.364, 0.06).
That we did in the video.
For the quiz you should have calculated these values, (-0.614, +0.4092)
and computers being what they are, just keep doing this again and again and
produce this list of numbers.
Let's now look at this xy plot. So this is what I was describing a moment ago in the
video. Instead of plotting x vs. time and y vs. time, we can plot x vs y.
So we lose the time coordinate but we can see things two dimensionally.
The first value, our initial condition was (0.2, 0.4). So we started at x=0.2 and
y=0.4 and that's that point right there.
Then the next value in our itinerary was x=1.364 and a y=0.06 and that's this point
here. So this was our first point and that was our second. Our third point, the one
you calculated in the quiz was x_2=-0.614, so that's about over here, -0.61, and the
y value was 0.409 and there it is, that's the second iterate.
This is the initial condition, (x_0, y_0), this is (x_1, y_1), this is (x_2, y_2).
And then we can keep going trying to track the locations of (x_3, y_3), the fourth
iterate, the fifth iterate and so on.
So we see this dot jumps around the plane.
Often in dynamical systems we're interested just in the long term behavior
of the system. So it kind of looks like this is going to something periodic.
Let's test that out. Maybe I'll plot 100 iterates. Make the time series plot.
And it does indeed look like it's becoming almost, maybe it's period 2, maybe it's
period 4. It's a little bit hard to tell. Let's go back and look at this.
So here, this is the xy view. There is our starting point and there is the next point
there is the next point. And we can see that points are accumulating in here.
If we're interested in the long term behavior I don't want to plot these first
dozen or first 20 or 30 points. I'm really interested in just the points after the
map has been going for a while.
So that's where this box comes in. So when making the xy plot, this will let
me skip some of the iterates. I'm going to choose to skip the first 50 iterates.
So then what the xy plot will do, that's the blue plot down below, is it will, the
computer will iterate the function for 50 times and it won't plot the points on the
xy plot, but it'll only start plotting with the 51st iterate and should do the
next 50. So let's make plots.
The purple plots don't change.
But down here, you can see that our starting point, which was (0.2, 0.4) right
around here, that that seems to be gone. And we're starting to see that the points
are oscillating, more or less, between these two regions.
So we can't see the oscillation on this because we can't see time value.
But we can tell from here, that the orbits are oscillating between a high and a low,
high and low, but they're not quite exactly the same.
So let's see here. I'm going to now plot, or have it do, let's have it do 10,000
iterates, I better do 5,000, I don't want to use things too much. And I'm going to
have it skip plotting the first 4,900. So the purple plots, the time series
plots are always going to show all of the iterates. So let's get this started.
So we expect that to be a big, purple blob. We've seen those blobs before.
It's taking a little while. There it is.
So that's a time series plot of 5,000 iterates for x, there's y, but let's see
what this looks like.
So at this point, it's becoming clear that it looks like the function is resolving,
the behavior is resolving to where it's oscillating just between two points,
or maybe among four points, but the two ones are almost identical.
In any event, this is clearly some sort of a periodic behavior.
Maybe period 2, maybe period 4.
But the main thing is that the long term behavior is simple and it's periodic.
So for the logistic map we did a final state diagram for a periodic orbit and it
just had 2 or 4 or 8, however many points it was, corresponding to the periodicity
of the orbit. We see the same thing here but now the points live on a plane and not
on a line.
So we'll do a lot more with the Hénon map in the next unit where we'll see what
chaotic behavior in two and three dimensions looks like and we'll see
strange attractors. But for now, the main new thing here is that we have a map that
has an x and y in it and those x and y values get mixed up because in this case,
the x value depends on y and x. And we can plot two time series for that and
that's pretty familiar.
Let me go back here and get a picture that is going it to look better.
So we can plot time series like we have before but the new thing is that we can
plot, have those orbits evolve on a two dimensional grid, a two dimensional
plane, instead of a one dimensional one.