1
00:00:00 --> 00:00:06
2
00:00:00 --> 00:00:06
As a matter of fact,
it plots them very accurately.
3
00:00:03 --> 00:00:09
But it is something you also
need to learn to do yourself,
4
00:00:08 --> 00:00:14
as you will see when we study
nonlinear equations.
5
00:00:11 --> 00:00:17
It is a skill.
And since a couple of important
6
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mathematical ideas are involved
in it, I think it is a very good
7
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thing to spend just a little
time on, one lecture in fact,
8
00:00:24 --> 00:00:30
plus a little more on the
problem set that I will give
9
00:00:28 --> 00:00:34
out.
The last problem set that I
10
00:00:32 --> 00:00:38
will give out on Friday.
I thought it might be a little
11
00:00:36 --> 00:00:42
more fun to, again,
have a simple-minded model.
12
00:00:40 --> 00:00:46
No romance this time.
We are going to have a little
13
00:00:45 --> 00:00:51
model of war,
but I have made it sort of
14
00:00:48 --> 00:00:54
sublimated war.
Let's take as the system,
15
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I am going to let two of those
be parameters,
16
00:00:55 --> 00:01:01
you know, be variable,
in other words.
17
00:01:00 --> 00:01:06
And the other two I will keep
fixed, so that you can
18
00:01:05 --> 00:01:11
concentrate on them better.
I will take a and d to be
19
00:01:10 --> 00:01:16
negative 1 and negative 3.
And the other ones we will
20
00:01:15 --> 00:01:21
leave open, so let's call this
one b times y,
21
00:01:19 --> 00:01:25
and this other one will be c
times x.
22
00:01:24 --> 00:01:30
23
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I am going to model this as a
fight between two states,
24
00:01:43 --> 00:01:49
both of which are trying to
attract tourists.
25
00:01:50 --> 00:01:56
Let's say this is Massachusetts
and this will be New Hampshire,
26
00:01:58 --> 00:02:04
its enemy to the North.
Both are busy advertising these
27
00:02:05 --> 00:02:11
days on television.
People are making their summer
28
00:02:08 --> 00:02:14
plans.
Come to New Hampshire,
29
00:02:10 --> 00:02:16
you know, New Hampshire has
mountains and Massachusetts has
30
00:02:14 --> 00:02:20
quaint little fishing villages
and stuff like that.
31
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32
00:02:24 --> 00:02:30
So what are these numbers?
Well, first of all,
33
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what do x and y represent?
x and y basically are the
34
00:02:33 --> 00:02:39
advertising budgets for tourism,
you know, the amount each state
35
00:02:39 --> 00:02:45
plans to spend during the year.
However, I do not want zero
36
00:02:45 --> 00:02:51
value to mean they are not
spending anything.
37
00:02:49 --> 00:02:55
It represents departure from
the normal equilibrium.
38
00:02:54 --> 00:03:00
x and y represent departures --
39
00:02:59 --> 00:03:05
40
00:03:04 --> 00:03:10
-- from the normal amount of
money they spend advertising for
41
00:03:09 --> 00:03:15
tourists.
The normal tourist advertising
42
00:03:13 --> 00:03:19
budget.
43
00:03:14 --> 00:03:20
44
00:03:20 --> 00:03:26
If they are both zero,
it means that both states are
45
00:03:23 --> 00:03:29
spending what they normally
spend in that year.
46
00:03:26 --> 00:03:32
If x is positive,
it means that Massachusetts has
47
00:03:29 --> 00:03:35
decided to spend more in the
hope of attracting more tourists
48
00:03:33 --> 00:03:39
and if negative spending less.
What is the significance of
49
00:03:37 --> 00:03:43
these two coefficients?
Those are the normal things
50
00:03:41 --> 00:03:47
which return you to equilibrium.
In other words,
51
00:03:44 --> 00:03:50
if x gets bigger than normal,
if Massachusetts spends more
52
00:03:48 --> 00:03:54
there is a certain poll to spend
less because we are wasting all
53
00:03:53 --> 00:03:59
this money on the tourists that
are not going to come when we
54
00:03:57 --> 00:04:03
could be spending it on
education or something like
55
00:04:00 --> 00:04:06
that.
If x gets to be negative,
56
00:04:03 --> 00:04:09
the governor tries to spend
less.
57
00:04:05 --> 00:04:11
Then all the local city Chamber
of Commerce rise up and start
58
00:04:09 --> 00:04:15
screaming that our economy is
going to go bankrupt because we
59
00:04:13 --> 00:04:19
won't get enough tourists and
that is because you are not
60
00:04:16 --> 00:04:22
spending enough money.
There is a push to always
61
00:04:19 --> 00:04:25
return it to the normal,
and that is what this negative
62
00:04:22 --> 00:04:28
sign means.
The same thing for New
63
00:04:24 --> 00:04:30
Hampshire.
What does it mean that this is
64
00:04:27 --> 00:04:33
negative three and that is
negative one?
65
00:04:31 --> 00:04:37
It just means that the Chamber
of Commerce yells three times as
66
00:04:36 --> 00:04:42
loudly in New Hampshire.
It is more sensitive,
67
00:04:40 --> 00:04:46
in other words,
to changes in the budget.
68
00:04:44 --> 00:04:50
Now, how about the other?
Well, these represent the
69
00:04:48 --> 00:04:54
war-like features of the
situation.
70
00:04:51 --> 00:04:57
Normally these will be positive
numbers.
71
00:04:56 --> 00:05:02
Because when Massachusetts sees
that New Hampshire has budgeted
72
00:05:00 --> 00:05:06
this year more than its normal
amount, the natural instinct is
73
00:05:05 --> 00:05:11
we are fighting.
This is war.
74
00:05:08 --> 00:05:14
This is a positive number.
We have to budget more,
75
00:05:12 --> 00:05:18
too.
And the same thing for New
76
00:05:14 --> 00:05:20
Hampshire.
The size of these coefficients
77
00:05:17 --> 00:05:23
gives you the magnitude of the
reaction.
78
00:05:20 --> 00:05:26
If they are small Massachusetts
say, well, they are spending
79
00:05:25 --> 00:05:31
more but we don't have to follow
them.
80
00:05:30 --> 00:05:36
We will bucket a little bit.
If it is a big number then oh,
81
00:05:34 --> 00:05:40
my God, heads will roll.
We have to triple them and put
82
00:05:38 --> 00:05:44
them out of business.
This is a model,
83
00:05:41 --> 00:05:47
in fact, for all sorts of
competition.
84
00:05:44 --> 00:05:50
It was used for many years to
model in simper times armaments
85
00:05:49 --> 00:05:55
races between countries.
It is certainly a simple-minded
86
00:05:53 --> 00:05:59
model for any two companies in
competition with each other if
87
00:05:58 --> 00:06:04
certain conditions are met.
Well, what I would like to do
88
00:06:05 --> 00:06:11
now is try different values of
those numbers.
89
00:06:10 --> 00:06:16
And, in each case,
show you how to sketch the
90
00:06:15 --> 00:06:21
solutions at different cases.
And then, for each different
91
00:06:22 --> 00:06:28
case, we will try to interpret
if it makes sense or not.
92
00:06:30 --> 00:06:36
My first set of numbers is,
the first case is --
93
00:06:34 --> 00:06:40
94
00:06:43 --> 00:06:49
-- x prime equals negative x
plus 2y.
95
00:06:49 --> 00:06:55
And y prime equals,
this is going to be zero,
96
00:06:54 --> 00:07:00
so it is simply minus
3 times y.
97
00:07:00 --> 00:07:06
Now, what does this mean?
Well, this means that
98
00:07:04 --> 00:07:10
Massachusetts is behaving
normally, but New Hampshire is a
99
00:07:08 --> 00:07:14
very placid state,
and the governor is busy doing
100
00:07:13 --> 00:07:19
other things.
And people say Massachusetts is
101
00:07:17 --> 00:07:23
spending more this year,
and the Governor says,
102
00:07:21 --> 00:07:27
so what.
The zero is the so what factor.
103
00:07:24 --> 00:07:30
In other words,
we are not going to respond to
104
00:07:28 --> 00:07:34
them.
We will do our own thing.
105
00:07:31 --> 00:07:37
What is the result of this?
Is Massachusetts going to win
106
00:07:35 --> 00:07:41
out?
What is going to be the
107
00:07:37 --> 00:07:43
ultimate effect on the budget?
Well, what we have to do is,
108
00:07:40 --> 00:07:46
so the program is first let's
quickly solve the equations
109
00:07:44 --> 00:07:50
using a standard technique.
I am just going to make marks
110
00:07:48 --> 00:07:54
on the board and trust to the
fact that you have done enough
111
00:07:52 --> 00:07:58
of this yourself by now that you
know what the marks mean.
112
00:07:57 --> 00:08:03
I am not going to label what
everything is.
113
00:08:00 --> 00:08:06
I am just going to trust to
luck.
114
00:08:03 --> 00:08:09
The matrix A is negative 1,
2, 0, negative 3.
115
00:08:07 --> 00:08:13
The characteristic equation,
116
00:08:11 --> 00:08:17
the second coefficient is the
trace, which is minus 4,
117
00:08:15 --> 00:08:21
but you have to change its
sign, so that makes it plus 4.
118
00:08:20 --> 00:08:26
And the constant term is the
determinant, which is 3 minus 0,
119
00:08:25 --> 00:08:31
so that is plus 3 equals zero.
This factors into lambda plus 3
120
00:08:31 --> 00:08:37
times lambda plus one.
And it means the roots
121
00:08:35 --> 00:08:41
therefore are,
one root is lambda equals
122
00:08:38 --> 00:08:44
negative 3 and the other root is
lambda equals negative 1.
123
00:08:43 --> 00:08:49
These are the eigenvalues.
With each eigenvalue goes an
124
00:08:47 --> 00:08:53
eigenvector.
The eigenvector is found by
125
00:08:51 --> 00:08:57
solving an equation for the
coefficients of the eigenvector,
126
00:08:56 --> 00:09:02
the components of the
eigenvector.
127
00:09:00 --> 00:09:06
Here I used negative 1 minus
negative 3, which makes 2.
128
00:09:04 --> 00:09:10
The first equation is 2a1 plus
2a2 is equal to zero.
129
00:09:09 --> 00:09:15
The second one will be,
in fact, in this case simply
130
00:09:14 --> 00:09:20
0a1 plus 0a2 so it won't give me
any information at all.
131
00:09:18 --> 00:09:24
That is not what usually
happens, but it is what happens
132
00:09:23 --> 00:09:29
in this case.
What is the solution?
133
00:09:28 --> 00:09:34
The solution is the vector
alpha equals,
134
00:09:32 --> 00:09:38
well, 1, negative 1
would be a good thing to
135
00:09:37 --> 00:09:43
use.
That is the eigenvector,
136
00:09:40 --> 00:09:46
so this is the e-vector.
How about lambda equals
137
00:09:45 --> 00:09:51
negative 1?
Let's give it a little more
138
00:09:49 --> 00:09:55
room.
If lambda is negative 1 then
139
00:09:52 --> 00:09:58
here I put negative 1 minus
negative 1.
140
00:09:56 --> 00:10:02
That makes zero.
I will write in the zero
141
00:10:01 --> 00:10:07
because this is confusing.
It is zero times a1.
142
00:10:04 --> 00:10:10
And the next coefficient is 2
a2, is zero.
143
00:10:07 --> 00:10:13
People sometimes go bananas
over this, in spite of the fact
144
00:10:11 --> 00:10:17
that this is the easiest
possible case you can get.
145
00:10:15 --> 00:10:21
I guess if they go bananas over
it, it proves it is not all that
146
00:10:19 --> 00:10:25
easy, but it is easy.
What now is the eigenvector
147
00:10:22 --> 00:10:28
that goes with this?
Well, this term isn't there.
148
00:10:26 --> 00:10:32
It is zero.
The equation says that a2 has
149
00:10:30 --> 00:10:36
to be zero.
And it doesn't say anything
150
00:10:32 --> 00:10:38
about a1, so let's make it 1.
151
00:10:35 --> 00:10:41
152
00:10:40 --> 00:10:46
Now, out of this data,
the final step is to make the
153
00:10:44 --> 00:10:50
general solution.
What is it?
154
00:10:47 --> 00:10:53
(x, y) equals,
well, a constant times the
155
00:10:51 --> 00:10:57
first normal mode.
The solution constructed from
156
00:10:55 --> 00:11:01
the eigenvalue and the
eigenvector.
157
00:11:00 --> 00:11:06
That is going to be 1,
negative 1 e to the minus 3t.
158
00:11:04 --> 00:11:10
And then the other normal mode
159
00:11:08 --> 00:11:14
times an arbitrary constant will
be (1, 0) times e to the
160
00:11:12 --> 00:11:18
negative t.
161
00:11:14 --> 00:11:20
The lambda is this factor which
produces that,
162
00:11:18 --> 00:11:24
of course.
Now, one way of looking at it
163
00:11:21 --> 00:11:27
is, first of all,
get clearly in your head this
164
00:11:25 --> 00:11:31
is a pair of parametric
equations just like what you
165
00:11:29 --> 00:11:35
studied in 18.02.
Let's write them out explicitly
166
00:11:34 --> 00:11:40
just this once.
x equals c1 times e to the
167
00:11:38 --> 00:11:44
negative 3t plus c2 times e to
the negative t.
168
00:11:44 --> 00:11:50
And what is y?
169
00:11:47 --> 00:11:53
y is equal to minus c1 e to the
minus 3t plus zero.
170
00:11:52 --> 00:11:58
I can stop there.
171
00:11:55 --> 00:12:01
In some sense,
all I am asking you to do is
172
00:11:59 --> 00:12:05
plot that curve.
In the x,y-plane,
173
00:12:03 --> 00:12:09
plot the curve given by this
pair of parametric equations.
174
00:12:07 --> 00:12:13
And you can choose your own
values of c1,
175
00:12:10 --> 00:12:16
c2.
For different values of c1 and
176
00:12:12 --> 00:12:18
c2 there will be different
curves.
177
00:12:15 --> 00:12:21
Give me a feeling for what they
all look like.
178
00:12:18 --> 00:12:24
Well, I think most of you will
recognize you didn't have stuff
179
00:12:22 --> 00:12:28
like this.
These weren't the kind of
180
00:12:25 --> 00:12:31
curves you plotted.
When you did parametric
181
00:12:29 --> 00:12:35
equations in 18.02,
you did stuff like x equals
182
00:12:32 --> 00:12:38
cosine t, y equals sine t.
183
00:12:36 --> 00:12:42
Everybody knows how to do that.
A few other curves which made
184
00:12:40 --> 00:12:46
lines or nice things,
but nothing that ever looked
185
00:12:43 --> 00:12:49
like that.
And so the computer will plot
186
00:12:45 --> 00:12:51
it by actually calculating
values but, of course,
187
00:12:48 --> 00:12:54
we will not.
That is the significance of the
188
00:12:51 --> 00:12:57
word sketch.
I am not asking you to plot
189
00:12:54 --> 00:13:00
carefully, but to give me some
general geometric picture of
190
00:12:58 --> 00:13:04
what all these curves look like
without doing any work.
191
00:13:03 --> 00:13:09
Without doing any work.
Well, that sounds promising.
192
00:13:09 --> 00:13:15
Okay, let's try to do it
without doing any work.
193
00:13:16 --> 00:13:22
Where shall I begin?
Hidden in this formula are four
194
00:13:23 --> 00:13:29
solutions that are extremely
easy to plot.
195
00:13:30 --> 00:13:36
So begin with the four easy
solutions, and then fill in the
196
00:13:38 --> 00:13:44
rest.
Now, which are the easy
197
00:13:42 --> 00:13:48
solutions?
The easy solutions are c1
198
00:13:47 --> 00:13:53
equals plus or minus 1,
c2 equals zero,
199
00:13:53 --> 00:13:59
or c1 equals zero,
or c1 = 0, c2 equals plus or
200
00:14:00 --> 00:14:06
minus 1.
By choosing those four values
201
00:14:05 --> 00:14:11
of c1 and c2,
I get simple solutions
202
00:14:07 --> 00:14:13
corresponding to the normal
mode.
203
00:14:10 --> 00:14:16
If c1 is one and c2 is zero,
I am talking about (1,
204
00:14:14 --> 00:14:20
negative 1) e to the minus 3t,
205
00:14:17 --> 00:14:23
and that is very easy plot.
Let's start plotting them.
206
00:14:21 --> 00:14:27
What I am going to do is
color-code them so you will be
207
00:14:25 --> 00:14:31
able to recognize what it is I
am plotting.
208
00:14:30 --> 00:14:36
Let's see.
What colors should we use?
209
00:14:33 --> 00:14:39
We will use pink and orange.
This will be our pink solution
210
00:14:39 --> 00:14:45
and our orange solution will be
this one.
211
00:14:43 --> 00:14:49
Let's plot the pink solution
first.
212
00:14:47 --> 00:14:53
The pink solution corresponds
to c1 equals 1 and c2
213
00:14:53 --> 00:14:59
equals zero.
Now, that solution looks like--
214
00:14:58 --> 00:15:04
Let's write it in pink.
215
00:15:01 --> 00:15:07
No, let's not write it in pink.
What is the solution?
216
00:15:06 --> 00:15:12
It looks like x equals e to the
negative 3t,
217
00:15:11 --> 00:15:17
y equals minus e to the minus
3t.
218
00:15:15 --> 00:15:21
Well, that's not a good way to
look at it, actually.
219
00:15:19 --> 00:15:25
The best way to look at it is
to say at t equals zero,
220
00:15:24 --> 00:15:30
where is it?
It is at the point 1,
221
00:15:26 --> 00:15:32
negative 1.
222
00:15:30 --> 00:15:36
And what is it doing as t
increases?
223
00:15:32 --> 00:15:38
Well, it keeps the direction,
but travels.
224
00:15:35 --> 00:15:41
The amplitude,
the distance from the origin
225
00:15:39 --> 00:15:45
keeps shrinking.
As t increases,
226
00:15:41 --> 00:15:47
this factor,
so it is the tip of this
227
00:15:44 --> 00:15:50
vector, except the vector is
shrinking.
228
00:15:47 --> 00:15:53
It is still in the direction of
1, negative 1,
229
00:15:51 --> 00:15:57
but it is shrinking in
length because its amplitude is
230
00:15:55 --> 00:16:01
shrinking according to the law e
to the negative 3t.
231
00:16:02 --> 00:16:08
In other words,
this curve looks like this.
232
00:16:05 --> 00:16:11
At t equals zero it is over
here, and it goes along this
233
00:16:09 --> 00:16:15
diagonal line until as t equals
infinity, it gets to infinity,
234
00:16:14 --> 00:16:20
it reaches the origin.
Of course, it never gets there.
235
00:16:18 --> 00:16:24
It goes slower and slower and
slower in order that it may
236
00:16:23 --> 00:16:29
never reach the origin.
What was it doing for values of
237
00:16:27 --> 00:16:33
t less than zero?
The same thing,
238
00:16:31 --> 00:16:37
except it was further away.
It comes in from infinity along
239
00:16:35 --> 00:16:41
that straight line.
In other words,
240
00:16:37 --> 00:16:43
the eigenvector determines the
line on which it travels and the
241
00:16:41 --> 00:16:47
eigenvalue determines which way
it goes.
242
00:16:44 --> 00:16:50
If the eigenvalue is negative,
it is approaching the origin as
243
00:16:48 --> 00:16:54
t increases.
How about the other one?
244
00:16:51 --> 00:16:57
Well, if c1 is negative 1,
then everything is the
245
00:16:55 --> 00:17:01
same except it is the mirror
image of this one.
246
00:17:00 --> 00:17:06
If c1 is negative 1,
then at t equals zero it is at
247
00:17:03 --> 00:17:09
this point.
And, once again,
248
00:17:05 --> 00:17:11
the same reasoning shows that
it is coming into the origin as
249
00:17:10 --> 00:17:16
t increases.
I have now two solutions,
250
00:17:12 --> 00:17:18
this one corresponding to c1
equals 1,
251
00:17:16 --> 00:17:22
and the other one c2 equals
zero.
252
00:17:19 --> 00:17:25
This one corresponds to c1
equals negative 1.
253
00:17:22 --> 00:17:28
How about the other guy,
the orange guy?
254
00:17:25 --> 00:17:31
Well, now c1 is zero,
c2 is one, let's say.
255
00:17:30 --> 00:17:36
It is the vector (1,
0), but otherwise everything is
256
00:17:33 --> 00:17:39
the same.
I start now at the point (1,
257
00:17:36 --> 00:17:42
0) at time zero.
And, as t increases,
258
00:17:39 --> 00:17:45
I come into the origin always
along that direction.
259
00:17:42 --> 00:17:48
And before that I came in from
infinity.
260
00:17:45 --> 00:17:51
And, again, if c2 is 1
and if c2 is negative 1,
261
00:17:50 --> 00:17:56
I do the same thing but
on the other side.
262
00:17:55 --> 00:18:01
263
00:18:00 --> 00:18:06
That wasn't very hard.
I plotted four solutions.
264
00:18:04 --> 00:18:10
And now I roll up my sleeves
and waive my hands to try to get
265
00:18:10 --> 00:18:16
others.
The general philosophy is the
266
00:18:14 --> 00:18:20
following.
The general philosophy is the
267
00:18:18 --> 00:18:24
differential equation looks like
this.
268
00:18:21 --> 00:18:27
It is a system of differential
equations.
269
00:18:25 --> 00:18:31
These are continuous functions.
That means when I draw the
270
00:18:31 --> 00:18:37
velocity field corresponding to
that system of differential
271
00:18:36 --> 00:18:42
equations, because their
functions are continuous,
272
00:18:39 --> 00:18:45
as I move from one (x,
y) point to another the
273
00:18:43 --> 00:18:49
direction of the velocity
vectors change continuously.
274
00:18:46 --> 00:18:52
It never suddenly reverses
without something like that.
275
00:18:50 --> 00:18:56
Now, if that changes
continuously then the
276
00:18:53 --> 00:18:59
trajectories must change
continuously,
277
00:18:56 --> 00:19:02
too.
In other words,
278
00:18:59 --> 00:19:05
nearby trajectories should be
doing approximately the same
279
00:19:03 --> 00:19:09
thing.
Well, that means all the other
280
00:19:05 --> 00:19:11
trajectories are ones which come
like that must be going also
281
00:19:10 --> 00:19:16
toward the origin.
If I start here,
282
00:19:12 --> 00:19:18
probably I have to follow this
one.
283
00:19:15 --> 00:19:21
They are all coming to the
origin, but that is a little too
284
00:19:19 --> 00:19:25
vague.
How do they come to the origin?
285
00:19:22 --> 00:19:28
In other words,
are they coming in straight
286
00:19:25 --> 00:19:31
like that?
Probably not.
287
00:19:26 --> 00:19:32
Then what are they doing?
Now we are coming to the only
288
00:19:32 --> 00:19:38
point in the lecture which you
might find a little difficult.
289
00:19:36 --> 00:19:42
Try to follow what I am doing
now.
290
00:19:38 --> 00:19:44
If you don't follow,
it is not well done in the
291
00:19:42 --> 00:19:48
textbook, but it is very well
done in the notes because I
292
00:19:46 --> 00:19:52
wrote them myself.
Please, it is done very
293
00:19:49 --> 00:19:55
carefully in the notes,
patiently follow through the
294
00:19:52 --> 00:19:58
explanation.
It takes about that much space.
295
00:19:55 --> 00:20:01
It is one of the important
ideas that your engineering
296
00:19:59 --> 00:20:05
professors will expect you to
understand.
297
00:20:04 --> 00:20:10
Anyway, I know this only from
the negative one because they
298
00:20:08 --> 00:20:14
say to me at lunch,
ruin my lunch by saying I said
299
00:20:12 --> 00:20:18
it to my students and got
nothing but blank looks.
300
00:20:16 --> 00:20:22
What do you guys teach them
over there?
301
00:20:19 --> 00:20:25
Blah, blah, blah.
Maybe we ought to start
302
00:20:22 --> 00:20:28
teaching it ourselves.
Sure.
303
00:20:25 --> 00:20:31
Why don't they start cutting
their own hair,
304
00:20:28 --> 00:20:34
too?
305
00:20:30 --> 00:20:36
306
00:20:35 --> 00:20:41
Here is the idea.
Let me recopy that solution.
307
00:20:40 --> 00:20:46
The solution looks like (1,
negative 1) e to the minus 3t
308
00:20:46 --> 00:20:52
plus c2, (1, 0) e to the
negative t.
309
00:20:51 --> 00:20:57
310
00:20:56 --> 00:21:02
What I ask is as t goes to
infinity, I feel sure that the
311
00:21:00 --> 00:21:06
trajectories must be coming into
the origin because these guys
312
00:21:04 --> 00:21:10
are doing that.
And, in fact,
313
00:21:06 --> 00:21:12
that is confirmed.
As t goes to infinity,
314
00:21:09 --> 00:21:15
this goes to zero and that goes
to zero regardless of what the
315
00:21:13 --> 00:21:19
c1 and c2 are.
That makes it clear that this
316
00:21:17 --> 00:21:23
goes to zero no matter what the
c1 and c2 are as t goes to
317
00:21:21 --> 00:21:27
infinity, but I would like to
analyze it a little more
318
00:21:25 --> 00:21:31
carefully.
As t goes to infinity,
319
00:21:28 --> 00:21:34
I have the sum of two terms.
And what I ask is,
320
00:21:32 --> 00:21:38
which term is dominant?
Of these two terms,
321
00:21:36 --> 00:21:42
are they of equal importance,
or is one more important than
322
00:21:41 --> 00:21:47
the other?
When t is 10,
323
00:21:43 --> 00:21:49
for example,
that is not very far on the way
324
00:21:47 --> 00:21:53
to infinity, but it is certainly
far enough to illustrate.
325
00:21:52 --> 00:21:58
Well, e to the minus 10
is an extremely
326
00:21:56 --> 00:22:02
small number.
The only thing smaller is e to
327
00:22:01 --> 00:22:07
the minus 30.
The term that dominates,
328
00:22:05 --> 00:22:11
they are both small,
but relatively-speaking this
329
00:22:08 --> 00:22:14
one is much larger because this
one only has the factor e to the
330
00:22:13 --> 00:22:19
minus 10,
whereas, this has the factor e
331
00:22:17 --> 00:22:23
to the minus 30,
which is vanishingly small.
332
00:22:22 --> 00:22:28
In other words,
as t goes to infinity --
333
00:22:26 --> 00:22:32
Well, let's write it the other
way.
334
00:22:28 --> 00:22:34
This is the dominant term,
as t goes to infinity.
335
00:22:32 --> 00:22:38
336
00:22:38 --> 00:22:44
Now, just the opposite is true
as t goes to minus infinity.
337
00:22:43 --> 00:22:49
t going to minus infinity means
I am backing up along these
338
00:22:48 --> 00:22:54
curves.
As t goes to minus infinity,
339
00:22:51 --> 00:22:57
let's say t gets to be negative
100, this is e to the 100,
340
00:22:56 --> 00:23:02
but this is e to the 300,
341
00:23:01 --> 00:23:07
which is much,
much bigger.
342
00:23:03 --> 00:23:09
So this is the dominant term as
t goes to negative infinity.
343
00:23:10 --> 00:23:16
344
00:23:18 --> 00:23:24
Now what I have is the sum of
two vectors.
345
00:23:20 --> 00:23:26
Let's first look at what
happens as t goes to infinity.
346
00:23:24 --> 00:23:30
As t goes to infinity,
I have the sum of two vectors.
347
00:23:28 --> 00:23:34
This one is completely
negligible compared with the one
348
00:23:31 --> 00:23:37
on the right-hand side.
In other words,
349
00:23:35 --> 00:23:41
for a all intents and purposes,
as t goes to infinity,
350
00:23:38 --> 00:23:44
it is this thing that takes
over.
351
00:23:41 --> 00:23:47
Therefore, what does the
solution look like as t goes to
352
00:23:45 --> 00:23:51
infinity?
The answer is it follows the
353
00:23:47 --> 00:23:53
yellow line.
Now, what does it look like as
354
00:23:50 --> 00:23:56
it backs up?
As it came in from negative
355
00:23:53 --> 00:23:59
infinity, what does it look
like?
356
00:23:56 --> 00:24:02
Now, this one is a little
harder to see.
357
00:24:00 --> 00:24:06
This is big,
but this is infinity bigger.
358
00:24:03 --> 00:24:09
I mean very,
very much bigger,
359
00:24:06 --> 00:24:12
when t is a large negative
number.
360
00:24:09 --> 00:24:15
Therefore, what I have is the
sum of a very big vector.
361
00:24:14 --> 00:24:20
You're standing on the moon
looking at the blackboard,
362
00:24:19 --> 00:24:25
so this is really big.
This is a very big vector.
363
00:24:24 --> 00:24:30
This is one million meters
long, and this is only 20
364
00:24:29 --> 00:24:35
meters long.
That is this guy,
365
00:24:33 --> 00:24:39
and that is this guy.
I want the sum of those two.
366
00:24:36 --> 00:24:42
What does the sum look like?
The answer is a sum is
367
00:24:40 --> 00:24:46
approximately parallel to the
long guy because this is
368
00:24:44 --> 00:24:50
negligible.
This does not mean they are
369
00:24:47 --> 00:24:53
next to each other.
They are slightly tilted over,
370
00:24:51 --> 00:24:57
but not very much.
In other words,
371
00:24:53 --> 00:24:59
as t goes to negative infinity
it doesn't coincide with this
372
00:24:58 --> 00:25:04
vector.
The solution doesn't,
373
00:25:01 --> 00:25:07
but it is parallel to it.
It has the same direction.
374
00:25:05 --> 00:25:11
I am done.
It means far away from the
375
00:25:07 --> 00:25:13
origin, it should be parallel to
the pink line.
376
00:25:11 --> 00:25:17
Near the origin it should turn
and become more or less
377
00:25:15 --> 00:25:21
coincident with the orange line.
And those were the solutions.
378
00:25:19 --> 00:25:25
That's how they look.
379
00:25:22 --> 00:25:28
380
00:25:27 --> 00:25:33
How about down here?
The same thing,
381
00:25:30 --> 00:25:36
like that, but then after a
while they turn and join.
382
00:25:35 --> 00:25:41
Here, they have to turn around
to join up, but they join.
383
00:25:41 --> 00:25:47
And that is,
in a simple way,
384
00:25:44 --> 00:25:50
the sketches of those
functions.
385
00:25:47 --> 00:25:53
That is how they must look.
What does this say about our
386
00:25:53 --> 00:25:59
state?
Well, it says that the fact
387
00:25:57 --> 00:26:03
that the governor of New
Hampshire is indifferent to what
388
00:26:01 --> 00:26:07
Massachusetts is doing produces
ultimately harmony.
389
00:26:06 --> 00:26:12
Both states revert ultimately
their normal advertising budgets
390
00:26:11 --> 00:26:17
in spite of the fact that
Massachusetts is keeping an eye
391
00:26:15 --> 00:26:21
peeled out for the slightest
misbehavior on the part of New
392
00:26:20 --> 00:26:26
Hampshire.
Peace reins,
393
00:26:22 --> 00:26:28
in other words.
Now you should know some names.
394
00:26:27 --> 00:26:33
Let's see.
I will write names in purple.
395
00:26:30 --> 00:26:36
There are two words that are
used to describe this situation.
396
00:26:35 --> 00:26:41
First is the word that
describes the general pattern of
397
00:26:40 --> 00:26:46
the way these lines look.
The word for that is a node.
398
00:26:44 --> 00:26:50
And the fact that all the
trajectories end up at the
399
00:26:48 --> 00:26:54
origin for that one uses the
word sink.
400
00:26:52 --> 00:26:58
This could be modified to nodal
sink.
401
00:26:55 --> 00:27:01
That would be better.
Nodal sink, let's say.
402
00:27:00 --> 00:27:06
Nodal sink or,
if you like to write them in
403
00:27:03 --> 00:27:09
the opposite order,
sink node.
404
00:27:06 --> 00:27:12
In the same way there would be
something called a source node
405
00:27:11 --> 00:27:17
if I reversed all the arrows.
I am not going to calculate an
406
00:27:16 --> 00:27:22
example.
Why don't I simply do it by
407
00:27:19 --> 00:27:25
giving you --
For example,
408
00:27:23 --> 00:27:29
if the matrix A produced a
solution instead of that one.
409
00:27:28 --> 00:27:34
Suppose it looked like 1,
negative 1 e to the 3t.
410
00:27:32 --> 00:27:38
The eigenvalues were reversed,
411
00:27:36 --> 00:27:42
were now positive.
And I will make the other one
412
00:27:41 --> 00:27:47
positive, too.
c2 1, 0 e to the t.
413
00:27:44 --> 00:27:50
414
00:27:47 --> 00:27:53
415
00:27:57 --> 00:28:03
What would that change in the
picture?
416
00:27:59 --> 00:28:05
The answer is essentially
nothing, except the direction of
417
00:28:04 --> 00:28:10
the arrows.
In other words,
418
00:28:06 --> 00:28:12
the first thing would still be
1, negative 1.
419
00:28:09 --> 00:28:15
The only difference is that now
420
00:28:12 --> 00:28:18
as t increases we go the other
way.
421
00:28:15 --> 00:28:21
And here the same thing,
we have still the same basic
422
00:28:19 --> 00:28:25
vector, the same basic orange
vector, orange line,
423
00:28:22 --> 00:28:28
but it has now traversed the
solution.
424
00:28:25 --> 00:28:31
We traverse it in the opposite
direction.
425
00:28:30 --> 00:28:36
Now, let's do the same thing
about dominance,
426
00:28:35 --> 00:28:41
as we did before.
Which term dominates as t goes
427
00:28:40 --> 00:28:46
to infinity?
This is the dominant term.
428
00:28:44 --> 00:28:50
Because, as t goes to infinity,
3t is much bigger than t.
429
00:28:51 --> 00:28:57
This one, on the other hand,
dominates as t goes to negative
430
00:28:57 --> 00:29:03
infinity.
431
00:29:00 --> 00:29:06
432
00:29:05 --> 00:29:11
How now will the solutions look
like?
433
00:29:07 --> 00:29:13
Well, as t goes to infinity,
they follow the pink curve.
434
00:29:11 --> 00:29:17
Whereas, as t starts out from
negative infinity,
435
00:29:15 --> 00:29:21
they follow the orange curve.
436
00:29:18 --> 00:29:24
437
00:29:28 --> 00:29:34
As t goes to infinity,
they become parallel to the
438
00:29:33 --> 00:29:39
pink curve, and as t goes to
negative infinity,
439
00:29:38 --> 00:29:44
they are very close to the
origin and are following the
440
00:29:44 --> 00:29:50
yellow curve.
This is pink and this is
441
00:29:48 --> 00:29:54
yellow.
They look like this.
442
00:29:53 --> 00:29:59
443
00:30:03 --> 00:30:09
Notice the picture basically is
the same.
444
00:30:06 --> 00:30:12
It is the picture of a node.
All that has happened is the
445
00:30:11 --> 00:30:17
arrows are reversed.
And, therefore,
446
00:30:14 --> 00:30:20
this would be called a nodal
source.
447
00:30:17 --> 00:30:23
The word source and sink
correspond to what you learned
448
00:30:21 --> 00:30:27
in 18.02 and 8.02,
I hope, also,
449
00:30:24 --> 00:30:30
or you could call it a source
node.
450
00:30:27 --> 00:30:33
Both phrases are used,
depending on how you want to
451
00:30:31 --> 00:30:37
use it in a sentence.
And another word for this,
452
00:30:37 --> 00:30:43
this would be called unstable
because all of the solutions
453
00:30:41 --> 00:30:47
starting out from near the
origin ultimately end up
454
00:30:45 --> 00:30:51
infinitely far away from the
origin.
455
00:30:47 --> 00:30:53
This would be called stable.
In fact, it would be called
456
00:30:52 --> 00:30:58
asymptotically stable.
I don't like the word
457
00:30:55 --> 00:31:01
asymptotically,
but it has become standard in
458
00:30:58 --> 00:31:04
the literature.
And, more important,
459
00:31:02 --> 00:31:08
it is standard in your
textbook.
460
00:31:05 --> 00:31:11
And I don't like to fight with
a textbook.
461
00:31:08 --> 00:31:14
It just ends up confusing
everybody, including me.
462
00:31:12 --> 00:31:18
That is enough for nodes.
I would like to talk now about
463
00:31:16 --> 00:31:22
some of the other cases that can
occur because they lead to
464
00:31:21 --> 00:31:27
completely different pictures
that you should understand.
465
00:31:26 --> 00:31:32
Let's look at the case where
our governors behave a little
466
00:31:30 --> 00:31:36
more badly, a little more
combatively.
467
00:31:35 --> 00:31:41
468
00:31:40 --> 00:31:46
It is x prime equals negative x
as before,
469
00:31:46 --> 00:31:52
but this time a firm response
by Massachusetts to any sign of
470
00:31:52 --> 00:31:58
increased activity by
stockpiling of advertising
471
00:31:58 --> 00:32:04
budgets.
Here let's say New Hampshire
472
00:32:03 --> 00:32:09
now is even worse.
Five times, quintuple or
473
00:32:08 --> 00:32:14
whatever increase Massachusetts
makes, of course they don't have
474
00:32:15 --> 00:32:21
an income tax,
but they will manage.
475
00:32:19 --> 00:32:25
Minus 3y as before.
Let's again calculate quickly
476
00:32:24 --> 00:32:30
what the characteristic equation
is.
477
00:32:30 --> 00:32:36
Our matrix is now negative 1,
3, 5 and negative 3.
478
00:32:34 --> 00:32:40
The characteristic equation now
479
00:32:37 --> 00:32:43
is lambda squared.
What is that?
480
00:32:40 --> 00:32:46
Again, plus 4 lambda.
But now the determinant is 3
481
00:32:44 --> 00:32:50
minus 15 is negative 12.
482
00:32:48 --> 00:32:54
And this, because I prepared
very carefully,
483
00:32:52 --> 00:32:58
all eigenvalues are integers.
And so this factors into lambda
484
00:32:57 --> 00:33:03
plus 6 times lambda minus 2,
485
00:33:01 --> 00:33:07
does it not?
Yes.
486
00:33:04 --> 00:33:10
6 lambda minus 2 is four
lambda.
487
00:33:07 --> 00:33:13
Good.
What do we have?
488
00:33:10 --> 00:33:16
Well, first of all we have our
eigenvalue lambda,
489
00:33:15 --> 00:33:21
negative 6.
And the eigenvector that goes
490
00:33:19 --> 00:33:25
with that is minus 1.
This is negative 1 minus
491
00:33:24 --> 00:33:30
negative 6 which makes,
shut your eyes,
492
00:33:28 --> 00:33:34
5.
We have 5a1 plus 3a2 is zero.
493
00:33:32 --> 00:33:38
And the other equation,
494
00:33:35 --> 00:33:41
I hope it comes out to be
something similar.
495
00:33:38 --> 00:33:44
I didn't check.
I am hoping this is right.
496
00:33:42 --> 00:33:48
The eigenvector is,
okay, you have been taught to
497
00:33:46 --> 00:33:52
always make one of the 1,
forget about that.
498
00:33:49 --> 00:33:55
Just pick numbers that make it
come out right.
499
00:33:53 --> 00:33:59
I am going to make this one 3,
and then I will make this one
500
00:33:57 --> 00:34:03
negative 5.
As I say, I have a policy of
501
00:34:02 --> 00:34:08
integers only.
I am a number theorist at
502
00:34:06 --> 00:34:12
heart.
That is how I started out life
503
00:34:09 --> 00:34:15
anyway.
There we have data from which
504
00:34:12 --> 00:34:18
we can make one solution.
How about the other one?
505
00:34:17 --> 00:34:23
The other one will correspond
to the eigenvalue lambda equals
506
00:34:23 --> 00:34:29
2.
This time the equation is
507
00:34:25 --> 00:34:31
negative 1 minus 2 is negative
3.
508
00:34:30 --> 00:34:36
It is minus 3a1 plus 3a2 is
zero.
509
00:34:34 --> 00:34:40
And now the eigenvector is (1,
1).
510
00:34:37 --> 00:34:43
Now we are ready to draw
pictures.
511
00:34:40 --> 00:34:46
We are going to make this
similar analysis,
512
00:34:44 --> 00:34:50
but it will go faster now
because you have already had the
513
00:34:49 --> 00:34:55
experience of that.
First of all,
514
00:34:52 --> 00:34:58
what is our general solution?
It is going to be c1 times 3,
515
00:34:57 --> 00:35:03
negative 5 e to the minus 6t.
516
00:35:02 --> 00:35:08
And then the other normal mode
517
00:35:06 --> 00:35:12
times an arbitrary constant will
be 1, 1 times e to the 2t.
518
00:35:11 --> 00:35:17
519
00:35:12 --> 00:35:18
520
00:35:18 --> 00:35:24
I am going to use the same
strategy.
521
00:35:20 --> 00:35:26
We have our two normal modes
here, eigenvalue,
522
00:35:24 --> 00:35:30
eigenvector solutions from
which, by adjusting these
523
00:35:27 --> 00:35:33
constants, we can get our four
basic solutions.
524
00:35:32 --> 00:35:38
Those are going to look like,
let's draw a picture here.
525
00:35:37 --> 00:35:43
Again, I will color-code them.
Let's use pink again.
526
00:35:42 --> 00:35:48
The pink solution now starts at
3, negative 5.
527
00:35:47 --> 00:35:53
That is where it is when t is
528
00:35:50 --> 00:35:56
zero.
And, because of the coefficient
529
00:35:54 --> 00:36:00
minus 6 up there,
it is coming into the origin
530
00:35:58 --> 00:36:04
and looks like that.
And its mirror image,
531
00:36:03 --> 00:36:09
of course, does the same thing.
That is when c1 is negative
532
00:36:08 --> 00:36:14
one.
How about the orange guy?
533
00:36:10 --> 00:36:16
Well, when t is equal to zero,
it is at 1, 1.
534
00:36:14 --> 00:36:20
But what is it doing after
535
00:36:16 --> 00:36:22
that?
As t increases,
536
00:36:18 --> 00:36:24
it is getting further away from
the origin because the sign here
537
00:36:22 --> 00:36:28
is positive.
e to the 2t is
538
00:36:25 --> 00:36:31
increasing, it is not decreasing
anymore, so this guy is going
539
00:36:30 --> 00:36:36
out.
And its mirror image on the
540
00:36:35 --> 00:36:41
other side is doing the same
thing.
541
00:36:40 --> 00:36:46
Now all we have to do is fill
in the picture.
542
00:36:46 --> 00:36:52
Well, you fill it in by
continuity.
543
00:36:51 --> 00:36:57
Your nearby trajectories must
be doing what similar thing?
544
00:37:00 --> 00:37:06
If I start out very near the
pink guy, I should stay near the
545
00:37:04 --> 00:37:10
pink guy.
But as I get near the origin,
546
00:37:07 --> 00:37:13
I am also approaching the
orange guy.
547
00:37:09 --> 00:37:15
Well, there is no other
possibility other than that.
548
00:37:13 --> 00:37:19
If you are further away you
start turning a little sooner.
549
00:37:17 --> 00:37:23
I am just using an argument
from continuity to say the
550
00:37:21 --> 00:37:27
picture must be roughly filled
out this way.
551
00:37:24 --> 00:37:30
Maybe not exactly.
In fact, there are fine points.
552
00:37:29 --> 00:37:35
And I am going to ask you to do
one of them on Friday for the
553
00:37:32 --> 00:37:38
new problem set,
even before the exam,
554
00:37:35 --> 00:37:41
God forbid.
But I want you to get a little
555
00:37:37 --> 00:37:43
more experience working with
that linear phase portrait
556
00:37:41 --> 00:37:47
visual because it is,
I think, one of the best ones
557
00:37:44 --> 00:37:50
this semester.
You can learn a lot from it.
558
00:37:47 --> 00:37:53
Anyway, you are not done with
it, but I hope you have at least
559
00:37:51 --> 00:37:57
looked at it by now.
That is what the picture looks
560
00:37:54 --> 00:38:00
like.
First of all,
561
00:37:55 --> 00:38:01
what are we going to name this?
In other words,
562
00:38:00 --> 00:38:06
forget about the arrows.
If you just look at the general
563
00:38:05 --> 00:38:11
way those lines go,
where have you seen this
564
00:38:08 --> 00:38:14
before?
You saw this in 18.02.
565
00:38:11 --> 00:38:17
What was the topic?
You were plotting contour
566
00:38:15 --> 00:38:21
curves of functions,
were you not?
567
00:38:18 --> 00:38:24
What did you call contours
curves that formed that pattern?
568
00:38:23 --> 00:38:29
A saddle point.
You called this a saddle point
569
00:38:26 --> 00:38:32
because it was like the center
of a saddle.
570
00:38:32 --> 00:38:38
It is like a mountain pass.
Here you are going up the
571
00:38:35 --> 00:38:41
mountain, say,
and here you are going down,
572
00:38:37 --> 00:38:43
the way the contour line is
going down.
573
00:38:40 --> 00:38:46
And this is sort of a min and
max point.
574
00:38:42 --> 00:38:48
A maximum if you go in that
direction and a minimum if you
575
00:38:46 --> 00:38:52
go in that direction,
say.
576
00:38:48 --> 00:38:54
Without the arrows on it,
it is like a saddle point.
577
00:38:51 --> 00:38:57
And so the same word is used
here.
578
00:38:53 --> 00:38:59
It is called the saddle.
You don't say point in the same
579
00:38:56 --> 00:39:02
way you don't say a nodal point.
It is the whole picture,
580
00:39:01 --> 00:39:07
as it were, that is the saddle.
It is a saddle.
581
00:39:05 --> 00:39:11
There is the saddle.
This is where you sit.
582
00:39:08 --> 00:39:14
Now, should I call it a source
or a sink?
583
00:39:12 --> 00:39:18
I cannot call it either because
it is a sink along these lines,
584
00:39:16 --> 00:39:22
it is a source along those
lines and along the others,
585
00:39:21 --> 00:39:27
it starts out looking like a
sink and then turns around and
586
00:39:25 --> 00:39:31
starts acting like a source.
The word source and sink are
587
00:39:31 --> 00:39:37
not used for saddle.
The only word that is used is
588
00:39:34 --> 00:39:40
unstable because definitely it
is unstable.
589
00:39:38 --> 00:39:44
If you start off exactly on the
pink lines you do end up at the
590
00:39:42 --> 00:39:48
origin, but if you start
anywhere else ever so close to a
591
00:39:47 --> 00:39:53
pink line you think you are
going to the origin,
592
00:39:50 --> 00:39:56
but then at the last minute you
are zooming off out to infinity
593
00:39:55 --> 00:40:01
again.
This is a typical example of
594
00:39:57 --> 00:40:03
instability.
Only if you do the
595
00:40:01 --> 00:40:07
mathematically possible,
but physically impossible thing
596
00:40:06 --> 00:40:12
of starting out exactly on the
pink line, only then will you
597
00:40:11 --> 00:40:17
get to the origin.
If you start out anywhere else,
598
00:40:15 --> 00:40:21
make the slightest error in
measure and get off the pink
599
00:40:20 --> 00:40:26
line, you end off at infinity.
What is the effect with our
600
00:40:25 --> 00:40:31
war-like governors fighting for
the tourist trade willing to
601
00:40:30 --> 00:40:36
spend any amounts of money to
match and overmatch what their
602
00:40:35 --> 00:40:41
competitor in the nearby state
is spending?
603
00:40:41 --> 00:40:47
The answer is,
they all lose.
604
00:40:43 --> 00:40:49
Since it is mostly this section
of the diagram that makes sense,
605
00:40:48 --> 00:40:54
what happens is they end up all
spending an infinity of dollars
606
00:40:53 --> 00:40:59
and nobody gets any more
tourists than anybody else.
607
00:40:58 --> 00:41:04
So this is a model of what not
to do.
608
00:41:02 --> 00:41:08
I have one more model to show
you.
609
00:41:05 --> 00:41:11
Maybe we better start over at
this board here.
610
00:41:11 --> 00:41:17
Massachusetts on top.
New Hampshire on the bottom.
611
00:41:17 --> 00:41:23
x prime is going to be,
that is Massachusetts,
612
00:41:23 --> 00:41:29
I guess as before.
Let me get the numbers right.
613
00:41:30 --> 00:41:36
614
00:41:45 --> 00:41:51
Leave that out for a moment.
y prime is 2x minus 3y.
615
00:41:50 --> 00:41:56
New Hampshire behaves normally.
616
00:41:54 --> 00:42:00
It is ready to respond to
anything Massachusetts can put
617
00:41:59 --> 00:42:05
out.
But by itself,
618
00:42:01 --> 00:42:07
it really wants to bring its
budget to normal.
619
00:42:05 --> 00:42:11
Now, Massachusetts,
we have a Mormon governor now,
620
00:42:09 --> 00:42:15
I guess.
Imagine instead we have a
621
00:42:12 --> 00:42:18
Buddhist governor.
A Buddhist governor reacts as
622
00:42:16 --> 00:42:22
follows, minus y.
What does that mean?
623
00:42:20 --> 00:42:26
It means that when he sees New
Hampshire increasing the budget,
624
00:42:25 --> 00:42:31
his reaction is,
we will lower ours.
625
00:42:30 --> 00:42:36
We will show them love.
It looks suicidal,
626
00:42:34 --> 00:42:40
but what actually happens?
Well, our little program is
627
00:42:39 --> 00:42:45
over.
Our matrix a is negative 1,
628
00:42:42 --> 00:42:48
negative 1, 2,
negative 3.
629
00:42:46 --> 00:42:52
The characteristic equations is
630
00:42:50 --> 00:42:56
lambda squared plus 4 lambda.
631
00:42:55 --> 00:43:01
And now what is the other term?
3 minus negative 2 makes 5.
632
00:43:02 --> 00:43:08
This is not going to factor
because I tried it out and I
633
00:43:07 --> 00:43:13
know it is not going to factor.
We are going to get lambda
634
00:43:13 --> 00:43:19
equals, we will just use the
quadratic formula,
635
00:43:17 --> 00:43:23
negative 4 plus or minus the
square root of 16 minus 4 times
636
00:43:23 --> 00:43:29
5, that is 16 minus 20 or
negative 4 all divided by 2,
637
00:43:28 --> 00:43:34
which makes minus 2,
pull out the 4,
638
00:43:31 --> 00:43:37
that makes it a 2,
cancels this 2,
639
00:43:35 --> 00:43:41
minus 1 inside.
It is minus 2 plus or minus i.
640
00:43:40 --> 00:43:46
Complex solutions.
641
00:43:44 --> 00:43:50
What are we doing to do about
that?
642
00:43:47 --> 00:43:53
Well, you should rejoice when
you get this case and are asked
643
00:43:53 --> 00:43:59
to sketch it because,
even if you calculate the
644
00:43:58 --> 00:44:04
complex eigenvector and from
that take its real and imaginary
645
00:44:04 --> 00:44:10
parts of the complex solution,
in fact, you will not be able
646
00:44:10 --> 00:44:16
easily to sketch the answer
anyway.
647
00:44:15 --> 00:44:21
But let me show you what sort
of thing you can get and then I
648
00:44:18 --> 00:44:24
am going to wave my hands and
argue a little bit to try to
649
00:44:21 --> 00:44:27
indicate what it is that the
solution actually looks like.
650
00:44:24 --> 00:44:30
You are going to get something
that looks like --
651
00:44:28 --> 00:44:34
A typical real solution is
going to look like this.
652
00:44:31 --> 00:44:37
This is going to produce e to
the minus 2t times e
653
00:44:36 --> 00:44:42
to the i t.
e to the minus 2 plus i all
654
00:44:40 --> 00:44:46
times t.
This will be our exponential
655
00:44:44 --> 00:44:50
factor which is shrinking in
amplitude.
656
00:44:47 --> 00:44:53
This is going to give me sines
and cosines.
657
00:44:50 --> 00:44:56
When I separate out the
eigenvector into its real and
658
00:44:54 --> 00:45:00
imaginary parts,
it is going to look something
659
00:44:57 --> 00:45:03
like this.
a1, a2 times cosine t,
660
00:45:02 --> 00:45:08
that is from the e to the it
661
00:45:05 --> 00:45:11
part.
Then there will be a sine term.
662
00:45:08 --> 00:45:14
And all that is going to be
multiplied by the exponential
663
00:45:12 --> 00:45:18
factor e to the negative 2t.
664
00:45:16 --> 00:45:22
665
00:45:22 --> 00:45:28
That is just one normal mode.
It is going to be c1 times this
666
00:45:28 --> 00:45:34
plus c2 times something similar.
It doesn't matter exactly what
667
00:45:34 --> 00:45:40
it is because they are all going
to look the same.
668
00:45:37 --> 00:45:43
Namely, this is a shrinking
amplitude.
669
00:45:40 --> 00:45:46
I am not going to worry about
that.
670
00:45:42 --> 00:45:48
My real question is,
what does this look like?
671
00:45:45 --> 00:45:51
In other words,
as a pair of parametric
672
00:45:48 --> 00:45:54
equations, if x is equal to a1
cosine t plus b1 sine t
673
00:45:52 --> 00:45:58
and y is a2
cosine plus b2 sine,
674
00:45:56 --> 00:46:02
what does it look like?
675
00:46:01 --> 00:46:07
Well, what are its
characteristics?
676
00:46:03 --> 00:46:09
In the first place,
as a curve this part of it is
677
00:46:08 --> 00:46:14
bounded.
It stays within some large box
678
00:46:11 --> 00:46:17
because cosine and sine never
get bigger than one and never
679
00:46:16 --> 00:46:22
get smaller than minus one.
It is periodic.
680
00:46:20 --> 00:46:26
As t increases to t plus 2pi,
681
00:46:24 --> 00:46:30
it comes back to exactly the
same point it was at before.
682
00:46:30 --> 00:46:36
683
00:46:35 --> 00:46:41
We have a curve that is
repeating itself periodically,
684
00:46:38 --> 00:46:44
it does not go off to infinity.
And here is where I am waving
685
00:46:42 --> 00:46:48
my hands.
It satisfies an equation.
686
00:46:44 --> 00:46:50
Those of you who like to fool
around with mathematics a little
687
00:46:49 --> 00:46:55
bit, it is not difficult to show
this, but it satisfies an
688
00:46:52 --> 00:46:58
equation of the form A x squared
plus B y squared plus C xy
689
00:46:56 --> 00:47:02
equals D.
690
00:47:00 --> 00:47:06
All you have to do is figure
out what the coefficients A,
691
00:47:03 --> 00:47:09
B, C and D should be.
And the way to do it is,
692
00:47:06 --> 00:47:12
if you calculate the square of
x you are going to get cosine
693
00:47:10 --> 00:47:16
squared, sine squared and a
cosine sine term.
694
00:47:13 --> 00:47:19
You are going to get those same
three terms here and the same
695
00:47:17 --> 00:47:23
three terms here.
You just use undetermined
696
00:47:20 --> 00:47:26
coefficients,
set up a system of simultaneous
697
00:47:23 --> 00:47:29
equations and you will be able
to find the A,
698
00:47:26 --> 00:47:32
B, C and D that work.
I am looking for a curve that
699
00:47:31 --> 00:47:37
is bounded, keeps repeating its
values and that satisfies a
700
00:47:35 --> 00:47:41
quadratic equation which looks
like this.
701
00:47:38 --> 00:47:44
Well, an earlier generation
would know from high school,
702
00:47:42 --> 00:47:48
these curves are all conic
sections.
703
00:47:45 --> 00:47:51
The only curves that satisfy
equations like that are
704
00:47:48 --> 00:47:54
hyperbola, parabolas,
the conic sections in other
705
00:47:52 --> 00:47:58
words, and ellipses.
Circles are a special kind of
706
00:47:56 --> 00:48:02
ellipses.
There is a degenerate case.
707
00:48:00 --> 00:48:06
A pair of lines which can be
considered a degenerate
708
00:48:04 --> 00:48:10
hyperbola, if you want.
It is as much a hyperbola as a
709
00:48:08 --> 00:48:14
circle, as an ellipse say.
Which of these is it?
710
00:48:11 --> 00:48:17
Well, it must be those guys.
Those are the only guys that
711
00:48:16 --> 00:48:22
stay bounded and repeat
themselves periodically.
712
00:48:20 --> 00:48:26
The other guys don't do that.
These are ellipses.
713
00:48:23 --> 00:48:29
And, therefore,
what do they look like?
714
00:48:28 --> 00:48:34
Well, they must look like an
ellipse that is trying to be an
715
00:48:32 --> 00:48:38
ellipse, but each time it goes
around the point is pulled a
716
00:48:37 --> 00:48:43
little closer to the origin.
It must be doing this,
717
00:48:41 --> 00:48:47
in other words.
And such a point is called a
718
00:48:44 --> 00:48:50
spiral sink.
Again sink because,
719
00:48:47 --> 00:48:53
no matter where you start,
you will get a curve that
720
00:48:51 --> 00:48:57
spirals into the origin.
Spiral is self-explanatory.
721
00:48:55 --> 00:49:01
And the one thing I haven't
told you that you must read is
722
00:49:00 --> 00:49:06
how do you know that it goes
around counterclockwise and not
723
00:49:04 --> 00:49:10
clockwise?
Read clockwise or
724
00:49:08 --> 00:49:14
counterclockwise.
I will give you the answer in
725
00:49:12 --> 00:49:18
30 seconds, not for this
particular curve.
726
00:49:16 --> 00:49:22
That you will have to
calculate.
727
00:49:19 --> 00:49:25
All you have to do is put in
somewhere.
728
00:49:23 --> 00:49:29
Let's say at the point (1,
0), a single vector from the
729
00:49:28 --> 00:49:34
velocity field.
In other words,
730
00:49:32 --> 00:49:38
at the point (1,
0), when x is 1 and y is 0 our
731
00:49:37 --> 00:49:43
vector is minus 1, 2,
732
00:49:40 --> 00:49:46
which is the vector minus 1,
2, it goes like this.
733
00:49:46 --> 00:49:52
Therefore, the motion must be
counterclockwise.
734
00:49:51 --> 00:49:57
And, by the way,
what is the effect of having a
735
00:49:56 --> 00:50:02
Buddhist governor?
Peace.
736
00:50:00 --> 00:50:06
Everything spirals into the
origin and everybody is left
737
00:50:05 --> 00:50:11
with the same advertising budget
they always had.
738
00:50:10 --> 00:50:16
Thanks.