1
00:00:02,790 --> 00:00:07,683
So we can iterate the Hénon map very
similar to how we iterated the logistic
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00:00:07,683 --> 00:00:13,131
equation. We just start with a value, in
this case it's two values and we just get
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the next value by applying the function
over and over again.
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00:00:16,257 --> 00:00:21,330
And we get a time series for x and a time
series for y.
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It's a straightforward task, soon we'll
turn to a computer to do this for us.
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So now that we have these numbers, what
can we do?
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Well, as usual in this course, we'd like
to know what's the long term behavior of
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00:00:34,180 --> 00:00:38,238
the orbit? Are there fixed points? Are
they stable? Can we have chaos? What does
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00:00:38,238 --> 00:00:39,322
it look like?
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00:00:40,278 --> 00:00:44,338
So we can plot the time series for x just
like we did for the logistic equation.
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00:00:44,338 --> 00:00:50,368
And we can plot the time series for y as
well. Those will both be one dimensional
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00:00:50,368 --> 00:00:54,835
functions, i.e. a dimension for x, a
dimension for y.
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00:00:56,478 --> 00:01:01,417
But another thing we can do, is plot x
against y. Similar to how we plotted x
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00:01:01,417 --> 00:01:06,060
against y or r against f for these two
dimensional differential equations.
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00:01:08,057 --> 00:01:13,117
So for the logistic equation we had a
final state diagram that was one
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00:01:13,117 --> 00:01:19,404
dimensional. Here, we can have a diagram
that would be two dimensional.
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00:01:25,062 --> 00:01:28,210
So I'll just sketch this and then we'll
look at it on the computer.
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00:01:29,746 --> 00:01:41,624
So the first point, x=0.2, y=0.4. I might
go over 2, up 4, maybe something sort of
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00:01:41,624 --> 00:01:49,683
roughly like that. And then the next point
x=1.3, so that might be maybe out here.
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00:01:50,510 --> 00:01:59,528
And y_1 is kind of down here. Because
x=1.364, y=0.06 so there's that point.
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00:02:00,615 --> 00:02:11,013
Then the next point now x is negative,
x=-0.6 and y=0.49, that looks like, let's
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see that would be probably be around here.
And let's see, this is point 0, and that
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00:02:20,871 --> 00:02:27,551
was point 1, this was point 2 and so on.
So I could keep plotting these dots, many,
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00:02:27,551 --> 00:02:33,848
many of them, and see what happens. See
if we get pulled into a fixed point or
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00:02:33,848 --> 00:02:36,931
maybe some sort of a cycle or something.
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00:02:36,931 --> 00:02:42,701
If I wanted to know just the long term
behavior, I might iterate for 10 or 100
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00:02:42,701 --> 00:02:47,953
times not plot those but then plot the
next 100 or something like that.
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00:02:48,418 --> 00:02:52,454
So this is very similar to what we did
for the final state diagram for the
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00:02:52,454 --> 00:02:57,477
logistic equation, but now instead of a
line we have a plane.
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00:02:57,477 --> 00:03:01,073
We have a plane because we have two
numbers, two coordinates, not just one.
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00:03:01,983 --> 00:03:07,441
So there's a program on the Complexity
Explorer site that will do this for us.
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00:03:08,021 --> 00:03:12,007
So let's take a look at that program now
and understand some of its features.
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00:03:18,272 --> 00:03:23,002
So here's the program on the Complexity
Explorer site that will make time series
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00:03:23,002 --> 00:03:29,447
plots of the Hénon map for us. You can
find this program on the site.
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00:03:29,447 --> 00:03:32,485
There's a link to it in the right hand
navigation bar.
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00:03:32,485 --> 00:03:39,127
I think it's section 7.6 or 7.7 that says
something like Hénon map programs.
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00:03:39,127 --> 00:03:41,029
Shouldn't be too hard to find.
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00:03:41,302 --> 00:03:45,884
So here's the program. It comes up in a
pop-up window that I've resized slightly.
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00:03:46,724 --> 00:03:52,216
It should look pretty familiar. It's very
similar to the program that we used to
40
00:03:52,216 --> 00:03:56,217
plot the logistic equation,
the logistic map.
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00:03:57,601 --> 00:04:02,191
So, first we can control the number of
iterations. I'll choose the default, 40.
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00:04:03,061 --> 00:04:11,161
Our initial conditions, the ones we've
been working with; x_0=0.2 but y_0=0.4.
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00:04:11,161 --> 00:04:20,751
So I'll change that. And the parameter a,
there's two parameters now, a=0.9, b=0.3.
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00:04:21,191 --> 00:04:27,968
And I'll describe what this box does in a
second. So let's make the plot.
45
00:04:30,190 --> 00:04:36,939
So this is a time series plot for the x
values; the x values change over time.
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00:04:36,939 --> 00:04:43,151
Time 0, time 1, time 2 and so on.
There's the time series plot.
47
00:04:43,151 --> 00:04:46,433
Here's a time series plot for the y value,
it also changes over time.
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00:04:46,433 --> 00:04:51,383
And we can make time series plots for both
x and y separately just like we did for
49
00:04:51,383 --> 00:04:52,499
the logistic equation.
50
00:04:54,885 --> 00:05:00,775
Let me slide down here.
It also produces a list of numbers.
51
00:05:00,775 --> 00:05:09,203
Here's our initial condition, (0.2, 0.4).
The first iterate was (1.364, 0.06).
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00:05:09,203 --> 00:05:11,158
That we did in the video.
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00:05:11,158 --> 00:05:19,258
For the quiz you should have calculated
these values, (-0.614, +0.4092)
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00:05:19,258 --> 00:05:24,310
and computers being what they are, just
keep doing this again and again and
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00:05:24,310 --> 00:05:28,208
produce this list of numbers.
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00:05:30,208 --> 00:05:37,332
Let's now look at this xy plot. So this is
what I was describing a moment ago in the
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00:05:37,332 --> 00:05:43,512
video. Instead of plotting x vs. time and
y vs. time, we can plot x vs y.
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00:05:43,512 --> 00:05:48,240
So we lose the time coordinate but we
can see things two dimensionally.
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00:05:55,980 --> 00:06:09,125
The first value, our initial condition was
(0.2, 0.4). So we started at x=0.2 and
60
00:06:09,125 --> 00:06:13,837
y=0.4 and that's that point right there.
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00:06:14,286 --> 00:06:26,359
Then the next value in our itinerary was
x=1.364 and a y=0.06 and that's this point
62
00:06:26,359 --> 00:06:36,451
here. So this was our first point and that
was our second. Our third point, the one
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00:06:36,451 --> 00:06:50,316
you calculated in the quiz was x_2=-0.614,
so that's about over here, -0.61, and the
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00:06:50,316 --> 00:06:55,928
y value was 0.409 and there it is, that's
the second iterate.
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00:06:55,928 --> 00:07:03,913
This is the initial condition, (x_0, y_0),
this is (x_1, y_1), this is (x_2, y_2).
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00:07:03,913 --> 00:07:11,738
And then we can keep going trying to track
the locations of (x_3, y_3), the fourth
67
00:07:11,738 --> 00:07:13,671
iterate, the fifth iterate and so on.
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00:07:14,371 --> 00:07:18,335
So we see this dot jumps around the plane.
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00:07:21,830 --> 00:07:25,478
Often in dynamical systems we're
interested just in the long term behavior
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00:07:25,478 --> 00:07:31,407
of the system. So it kind of looks like
this is going to something periodic.
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00:07:31,923 --> 00:07:40,200
Let's test that out. Maybe I'll plot 100
iterates. Make the time series plot.
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00:07:40,881 --> 00:07:45,094
And it does indeed look like it's becoming
almost, maybe it's period 2, maybe it's
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00:07:45,094 --> 00:07:54,836
period 4. It's a little bit hard to tell.
Let's go back and look at this.
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00:07:54,836 --> 00:08:03,045
So here, this is the xy view. There is our
starting point and there is the next point
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00:08:03,045 --> 00:08:07,710
there is the next point. And we can see
that points are accumulating in here.
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00:08:07,710 --> 00:08:12,640
If we're interested in the long term
behavior I don't want to plot these first
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00:08:12,640 --> 00:08:18,644
dozen or first 20 or 30 points. I'm really
interested in just the points after the
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00:08:18,644 --> 00:08:20,549
map has been going for a while.
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00:08:21,381 --> 00:08:25,444
So that's where this box comes in.
So when making the xy plot, this will let
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00:08:25,444 --> 00:08:31,723
me skip some of the iterates. I'm going to
choose to skip the first 50 iterates.
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00:08:32,973 --> 00:08:39,549
So then what the xy plot will do, that's
the blue plot down below, is it will, the
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00:08:39,549 --> 00:08:44,347
computer will iterate the function for 50
times and it won't plot the points on the
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00:08:44,347 --> 00:08:50,392
xy plot, but it'll only start plotting
with the 51st iterate and should do the
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00:08:50,392 --> 00:08:53,621
next 50. So let's make plots.
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00:08:54,189 --> 00:08:57,318
The purple plots don't change.
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00:08:57,318 --> 00:09:04,146
But down here, you can see that our
starting point, which was (0.2, 0.4) right
87
00:09:04,146 --> 00:09:08,600
around here, that that seems to be gone.
And we're starting to see that the points
88
00:09:08,600 --> 00:09:12,041
are oscillating, more or less, between
these two regions.
89
00:09:12,907 --> 00:09:18,285
So we can't see the oscillation on this
because we can't see time value.
90
00:09:18,285 --> 00:09:22,913
But we can tell from here, that the orbits
are oscillating between a high and a low,
91
00:09:22,913 --> 00:09:26,194
high and low, but they're not quite
exactly the same.
92
00:09:28,500 --> 00:09:36,487
So let's see here. I'm going to now plot,
or have it do, let's have it do 10,000
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00:09:36,487 --> 00:09:42,499
iterates, I better do 5,000, I don't want
to use things too much. And I'm going to
94
00:09:42,499 --> 00:09:48,458
have it skip plotting the first 4,900.
So the purple plots, the time series
95
00:09:48,458 --> 00:09:53,383
plots are always going to show all of
the iterates. So let's get this started.
96
00:09:54,065 --> 00:09:57,558
So we expect that to be a big, purple
blob. We've seen those blobs before.
97
00:09:58,697 --> 00:10:01,174
It's taking a little while. There it is.
98
00:10:04,123 --> 00:10:09,469
So that's a time series plot of 5,000
iterates for x, there's y, but let's see
99
00:10:09,469 --> 00:10:10,750
what this looks like.
100
00:10:11,848 --> 00:10:18,329
So at this point, it's becoming clear that
it looks like the function is resolving,
101
00:10:18,329 --> 00:10:22,674
the behavior is resolving to where it's
oscillating just between two points,
102
00:10:22,674 --> 00:10:26,838
or maybe among four points, but the two
ones are almost identical.
103
00:10:27,526 --> 00:10:30,605
In any event, this is clearly some sort of
a periodic behavior.
104
00:10:30,605 --> 00:10:32,782
Maybe period 2, maybe period 4.
105
00:10:34,282 --> 00:10:38,762
But the main thing is that the long term
behavior is simple and it's periodic.
106
00:10:39,496 --> 00:10:45,506
So for the logistic map we did a final
state diagram for a periodic orbit and it
107
00:10:45,506 --> 00:10:50,718
just had 2 or 4 or 8, however many points
it was, corresponding to the periodicity
108
00:10:50,718 --> 00:10:56,447
of the orbit. We see the same thing here
but now the points live on a plane and not
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00:10:56,447 --> 00:10:57,812
on a line.
110
00:11:00,010 --> 00:11:06,770
So we'll do a lot more with the Hénon map
in the next unit where we'll see what
111
00:11:06,770 --> 00:11:10,021
chaotic behavior in two and three
dimensions looks like and we'll see
112
00:11:10,021 --> 00:11:17,695
strange attractors. But for now, the main
new thing here is that we have a map that
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00:11:17,695 --> 00:11:23,241
has an x and y in it and those x and y
values get mixed up because in this case,
114
00:11:23,241 --> 00:11:29,312
the x value depends on y and x. And we
can plot two time series for that and
115
00:11:29,312 --> 00:11:31,074
that's pretty familiar.
116
00:11:31,074 --> 00:11:37,601
Let me go back here and get a picture
that is going it to look better.
117
00:11:37,601 --> 00:11:41,680
So we can plot time series like we have
before but the new thing is that we can
118
00:11:41,680 --> 00:11:45,222
plot, have those orbits evolve on a two
dimensional grid, a two dimensional
119
00:11:45,222 --> 00:11:48,838
plane, instead of a one dimensional one.