1
00:00:00,000 --> 00:00:06,000
As a matter of fact,
it plots them very accurately.

2
00:00:03,000 --> 00:00:09,000
But it is something you also
need to learn to do yourself,

3
00:00:08,000 --> 00:00:14,000
as you will see when we study
nonlinear equations.

4
00:00:11,000 --> 00:00:17,000
It is a skill.
And since a couple of important

5
00:00:15,000 --> 00:00:21,000
mathematical ideas are involved
in it, I think it is a very good

6
00:00:20,000 --> 00:00:26,000
thing to spend just a little
time on, one lecture in fact,

7
00:00:24,000 --> 00:00:30,000
plus a little more on the
problem set that I will give

8
00:00:28,000 --> 00:00:34,000
out.
The last problem set that I

9
00:00:32,000 --> 00:00:38,000
will give out on Friday.
I thought it might be a little

10
00:00:36,000 --> 00:00:42,000
more fun to, again,
have a simple-minded model.

11
00:00:40,000 --> 00:00:46,000
No romance this time.
We are going to have a little

12
00:00:45,000 --> 00:00:51,000
model of war,
but I have made it sort of

13
00:00:48,000 --> 00:00:54,000
sublimated war.
Let's take as the system,

14
00:00:51,000 --> 00:00:57,000
I am going to let two of those
be parameters,

15
00:00:55,000 --> 00:01:01,000
you know, be variable,
in other words.

16
00:01:00,000 --> 00:01:06,000
And the other two I will keep
fixed, so that you can

17
00:01:05,000 --> 00:01:11,000
concentrate on them better.
I will take a and d to be

18
00:01:10,000 --> 00:01:16,000
negative 1 and negative 3.
And the other ones we will

19
00:01:15,000 --> 00:01:21,000
leave open, so let's call this
one b times y,

20
00:01:19,000 --> 00:01:25,000
and this other one will be c
times x.

21
00:01:36,000 --> 00:01:42,000
I am going to model this as a
fight between two states,

22
00:01:43,000 --> 00:01:49,000
both of which are trying to
attract tourists.

23
00:01:50,000 --> 00:01:56,000
Let's say this is Massachusetts
and this will be New Hampshire,

24
00:01:58,000 --> 00:02:04,000
its enemy to the North.
Both are busy advertising these

25
00:02:05,000 --> 00:02:11,000
days on television.
People are making their summer

26
00:02:08,000 --> 00:02:14,000
plans.
Come to New Hampshire,

27
00:02:10,000 --> 00:02:16,000
you know, New Hampshire has
mountains and Massachusetts has

28
00:02:14,000 --> 00:02:20,000
quaint little fishing villages
and stuff like that.

29
00:02:24,000 --> 00:02:30,000
So what are these numbers?
Well, first of all,

30
00:02:28,000 --> 00:02:34,000
what do x and y represent?
x and y basically are the

31
00:02:33,000 --> 00:02:39,000
advertising budgets for tourism,
you know, the amount each state

32
00:02:39,000 --> 00:02:45,000
plans to spend during the year.
However, I do not want zero

33
00:02:45,000 --> 00:02:51,000
value to mean they are not
spending anything.

34
00:02:49,000 --> 00:02:55,000
It represents departure from
the normal equilibrium.

35
00:02:54,000 --> 00:03:00,000
x and y represent departures --

36
00:03:04,000 --> 00:03:10,000
-- from the normal amount of
money they spend advertising for

37
00:03:09,000 --> 00:03:15,000
tourists.
The normal tourist advertising

38
00:03:13,000 --> 00:03:19,000
budget.

39
00:03:20,000 --> 00:03:26,000
If they are both zero,
it means that both states are

40
00:03:23,000 --> 00:03:29,000
spending what they normally
spend in that year.

41
00:03:26,000 --> 00:03:32,000
If x is positive,
it means that Massachusetts has

42
00:03:29,000 --> 00:03:35,000
decided to spend more in the
hope of attracting more tourists

43
00:03:33,000 --> 00:03:39,000
and if negative spending less.
What is the significance of

44
00:03:37,000 --> 00:03:43,000
these two coefficients?
Those are the normal things

45
00:03:41,000 --> 00:03:47,000
which return you to equilibrium.
In other words,

46
00:03:44,000 --> 00:03:50,000
if x gets bigger than normal,
if Massachusetts spends more

47
00:03:48,000 --> 00:03:54,000
there is a certain poll to spend
less because we are wasting all

48
00:03:53,000 --> 00:03:59,000
this money on the tourists that
are not going to come when we

49
00:03:57,000 --> 00:04:03,000
could be spending it on
education or something like

50
00:04:00,000 --> 00:04:06,000
that.
If x gets to be negative,

51
00:04:03,000 --> 00:04:09,000
the governor tries to spend
less.

52
00:04:05,000 --> 00:04:11,000
Then all the local city Chamber
of Commerce rise up and start

53
00:04:09,000 --> 00:04:15,000
screaming that our economy is
going to go bankrupt because we

54
00:04:13,000 --> 00:04:19,000
won't get enough tourists and
that is because you are not

55
00:04:16,000 --> 00:04:22,000
spending enough money.
There is a push to always

56
00:04:19,000 --> 00:04:25,000
return it to the normal,
and that is what this negative

57
00:04:22,000 --> 00:04:28,000
sign means.
The same thing for New

58
00:04:24,000 --> 00:04:30,000
Hampshire.
What does it mean that this is

59
00:04:27,000 --> 00:04:33,000
negative three and that is
negative one?

60
00:04:31,000 --> 00:04:37,000
It just means that the Chamber
of Commerce yells three times as

61
00:04:36,000 --> 00:04:42,000
loudly in New Hampshire.
It is more sensitive,

62
00:04:40,000 --> 00:04:46,000
in other words,
to changes in the budget.

63
00:04:44,000 --> 00:04:50,000
Now, how about the other?
Well, these represent the

64
00:04:48,000 --> 00:04:54,000
war-like features of the
situation.

65
00:04:51,000 --> 00:04:57,000
Normally these will be positive
numbers.

66
00:04:56,000 --> 00:05:02,000
Because when Massachusetts sees
that New Hampshire has budgeted

67
00:05:00,000 --> 00:05:06,000
this year more than its normal
amount, the natural instinct is

68
00:05:05,000 --> 00:05:11,000
we are fighting.
This is war.

69
00:05:08,000 --> 00:05:14,000
This is a positive number.
We have to budget more,

70
00:05:12,000 --> 00:05:18,000
too.
And the same thing for New

71
00:05:14,000 --> 00:05:20,000
Hampshire.
The size of these coefficients

72
00:05:17,000 --> 00:05:23,000
gives you the magnitude of the
reaction.

73
00:05:20,000 --> 00:05:26,000
If they are small Massachusetts
say, well, they are spending

74
00:05:25,000 --> 00:05:31,000
more but we don't have to follow
them.

75
00:05:30,000 --> 00:05:36,000
We will bucket a little bit.
If it is a big number then oh,

76
00:05:34,000 --> 00:05:40,000
my God, heads will roll.
We have to triple them and put

77
00:05:38,000 --> 00:05:44,000
them out of business.
This is a model,

78
00:05:41,000 --> 00:05:47,000
in fact, for all sorts of
competition.

79
00:05:44,000 --> 00:05:50,000
It was used for many years to
model in simper times armaments

80
00:05:49,000 --> 00:05:55,000
races between countries.
It is certainly a simple-minded

81
00:05:53,000 --> 00:05:59,000
model for any two companies in
competition with each other if

82
00:05:58,000 --> 00:06:04,000
certain conditions are met.
Well, what I would like to do

83
00:06:05,000 --> 00:06:11,000
now is try different values of
those numbers.

84
00:06:10,000 --> 00:06:16,000
And, in each case,
show you how to sketch the

85
00:06:15,000 --> 00:06:21,000
solutions at different cases.
And then, for each different

86
00:06:22,000 --> 00:06:28,000
case, we will try to interpret
if it makes sense or not.

87
00:06:30,000 --> 00:06:36,000
My first set of numbers is,
the first case is --

88
00:06:43,000 --> 00:06:49,000
-- x prime equals negative x
plus 2y.

89
00:06:49,000 --> 00:06:55,000
And y prime equals,
this is going to be zero,

90
00:06:54,000 --> 00:07:00,000
so it is simply minus
3 times y.

91
00:07:00,000 --> 00:07:06,000
Now, what does this mean?
Well, this means that

92
00:07:04,000 --> 00:07:10,000
Massachusetts is behaving
normally, but New Hampshire is a

93
00:07:08,000 --> 00:07:14,000
very placid state,
and the governor is busy doing

94
00:07:13,000 --> 00:07:19,000
other things.
And people say Massachusetts is

95
00:07:17,000 --> 00:07:23,000
spending more this year,
and the Governor says,

96
00:07:21,000 --> 00:07:27,000
so what.
The zero is the so what factor.

97
00:07:24,000 --> 00:07:30,000
In other words,
we are not going to respond to

98
00:07:28,000 --> 00:07:34,000
them.
We will do our own thing.

99
00:07:31,000 --> 00:07:37,000
What is the result of this?
Is Massachusetts going to win

100
00:07:35,000 --> 00:07:41,000
out?
What is going to be the

101
00:07:37,000 --> 00:07:43,000
ultimate effect on the budget?
Well, what we have to do is,

102
00:07:40,000 --> 00:07:46,000
so the program is first let's
quickly solve the equations

103
00:07:44,000 --> 00:07:50,000
using a standard technique.
I am just going to make marks

104
00:07:48,000 --> 00:07:54,000
on the board and trust to the
fact that you have done enough

105
00:07:52,000 --> 00:07:58,000
of this yourself by now that you
know what the marks mean.

106
00:07:57,000 --> 00:08:03,000
I am not going to label what
everything is.

107
00:08:00,000 --> 00:08:06,000
I am just going to trust to
luck.

108
00:08:03,000 --> 00:08:09,000
The matrix A is negative 1,
2, 0, negative 3.

109
00:08:07,000 --> 00:08:13,000
The characteristic equation,

110
00:08:11,000 --> 00:08:17,000
the second coefficient is the
trace, which is minus 4,

111
00:08:15,000 --> 00:08:21,000
but you have to change its
sign, so that makes it plus 4.

112
00:08:20,000 --> 00:08:26,000
And the constant term is the
determinant, which is 3 minus 0,

113
00:08:25,000 --> 00:08:31,000
so that is plus 3 equals zero.
This factors into lambda plus 3

114
00:08:31,000 --> 00:08:37,000
times lambda plus one.
And it means the roots

115
00:08:35,000 --> 00:08:41,000
therefore are,
one root is lambda equals

116
00:08:38,000 --> 00:08:44,000
negative 3 and the other root is
lambda equals negative 1.

117
00:08:43,000 --> 00:08:49,000
These are the eigenvalues.
With each eigenvalue goes an

118
00:08:47,000 --> 00:08:53,000
eigenvector.
The eigenvector is found by

119
00:08:51,000 --> 00:08:57,000
solving an equation for the
coefficients of the eigenvector,

120
00:08:56,000 --> 00:09:02,000
the components of the
eigenvector.

121
00:09:00,000 --> 00:09:06,000
Here I used negative 1 minus
negative 3, which makes 2.

122
00:09:04,000 --> 00:09:10,000
The first equation is 2a1 plus
2a2 is equal to zero.

123
00:09:09,000 --> 00:09:15,000
The second one will be,
in fact, in this case simply

124
00:09:14,000 --> 00:09:20,000
0a1 plus 0a2 so it won't give me
any information at all.

125
00:09:18,000 --> 00:09:24,000
That is not what usually
happens, but it is what happens

126
00:09:23,000 --> 00:09:29,000
in this case.
What is the solution?

127
00:09:28,000 --> 00:09:34,000
The solution is the vector
alpha equals,

128
00:09:32,000 --> 00:09:38,000
well, 1, negative 1
would be a good thing to

129
00:09:37,000 --> 00:09:43,000
use.
That is the eigenvector,

130
00:09:40,000 --> 00:09:46,000
so this is the e-vector.
How about lambda equals

131
00:09:45,000 --> 00:09:51,000
negative 1?
Let's give it a little more

132
00:09:49,000 --> 00:09:55,000
room.
If lambda is negative 1 then

133
00:09:52,000 --> 00:09:58,000
here I put negative 1 minus
negative 1.

134
00:09:56,000 --> 00:10:02,000
That makes zero.
I will write in the zero

135
00:10:01,000 --> 00:10:07,000
because this is confusing.
It is zero times a1.

136
00:10:04,000 --> 00:10:10,000
And the next coefficient is 2
a2, is zero.

137
00:10:07,000 --> 00:10:13,000
People sometimes go bananas
over this, in spite of the fact

138
00:10:11,000 --> 00:10:17,000
that this is the easiest
possible case you can get.

139
00:10:15,000 --> 00:10:21,000
I guess if they go bananas over
it, it proves it is not all that

140
00:10:19,000 --> 00:10:25,000
easy, but it is easy.
What now is the eigenvector

141
00:10:22,000 --> 00:10:28,000
that goes with this?
Well, this term isn't there.

142
00:10:26,000 --> 00:10:32,000
It is zero.
The equation says that a2 has

143
00:10:30,000 --> 00:10:36,000
to be zero.
And it doesn't say anything

144
00:10:32,000 --> 00:10:38,000
about a1, so let's make it 1.

145
00:10:40,000 --> 00:10:46,000
Now, out of this data,
the final step is to make the

146
00:10:44,000 --> 00:10:50,000
general solution.
What is it?

147
00:10:47,000 --> 00:10:53,000
(x, y) equals,
well, a constant times the

148
00:10:51,000 --> 00:10:57,000
first normal mode.
The solution constructed from

149
00:10:55,000 --> 00:11:01,000
the eigenvalue and the
eigenvector.

150
00:11:00,000 --> 00:11:06,000
That is going to be 1,
negative 1 e to the minus 3t.

151
00:11:04,000 --> 00:11:10,000
And then the other normal mode

152
00:11:08,000 --> 00:11:14,000
times an arbitrary constant will
be (1, 0) times e to the

153
00:11:12,000 --> 00:11:18,000
negative t.

154
00:11:14,000 --> 00:11:20,000
The lambda is this factor which
produces that,

155
00:11:18,000 --> 00:11:24,000
of course.
Now, one way of looking at it

156
00:11:21,000 --> 00:11:27,000
is, first of all,
get clearly in your head this

157
00:11:25,000 --> 00:11:31,000
is a pair of parametric
equations just like what you

158
00:11:29,000 --> 00:11:35,000
studied in 18.02.
Let's write them out explicitly

159
00:11:34,000 --> 00:11:40,000
just this once.
x equals c1 times e to the

160
00:11:38,000 --> 00:11:44,000
negative 3t plus c2 times e to
the negative t.

161
00:11:44,000 --> 00:11:50,000
And what is y?

162
00:11:47,000 --> 00:11:53,000
y is equal to minus c1 e to the
minus 3t plus zero.

163
00:11:52,000 --> 00:11:58,000
I can stop there.

164
00:11:55,000 --> 00:12:01,000
In some sense,
all I am asking you to do is

165
00:11:59,000 --> 00:12:05,000
plot that curve.
In the x,y-plane,

166
00:12:03,000 --> 00:12:09,000
plot the curve given by this
pair of parametric equations.

167
00:12:07,000 --> 00:12:13,000
And you can choose your own
values of c1,

168
00:12:10,000 --> 00:12:16,000
c2.
For different values of c1 and

169
00:12:12,000 --> 00:12:18,000
c2 there will be different
curves.

170
00:12:15,000 --> 00:12:21,000
Give me a feeling for what they
all look like.

171
00:12:18,000 --> 00:12:24,000
Well, I think most of you will
recognize you didn't have stuff

172
00:12:22,000 --> 00:12:28,000
like this.
These weren't the kind of

173
00:12:25,000 --> 00:12:31,000
curves you plotted.
When you did parametric

174
00:12:29,000 --> 00:12:35,000
equations in 18.02,
you did stuff like x equals

175
00:12:32,000 --> 00:12:38,000
cosine t, y equals sine t.

176
00:12:36,000 --> 00:12:42,000
Everybody knows how to do that.
A few other curves which made

177
00:12:40,000 --> 00:12:46,000
lines or nice things,
but nothing that ever looked

178
00:12:43,000 --> 00:12:49,000
like that.
And so the computer will plot

179
00:12:45,000 --> 00:12:51,000
it by actually calculating
values but, of course,

180
00:12:48,000 --> 00:12:54,000
we will not.
That is the significance of the

181
00:12:51,000 --> 00:12:57,000
word sketch.
I am not asking you to plot

182
00:12:54,000 --> 00:13:00,000
carefully, but to give me some
general geometric picture of

183
00:12:58,000 --> 00:13:04,000
what all these curves look like
without doing any work.

184
00:13:03,000 --> 00:13:09,000
Without doing any work.
Well, that sounds promising.

185
00:13:09,000 --> 00:13:15,000
Okay, let's try to do it
without doing any work.

186
00:13:16,000 --> 00:13:22,000
Where shall I begin?
Hidden in this formula are four

187
00:13:23,000 --> 00:13:29,000
solutions that are extremely
easy to plot.

188
00:13:30,000 --> 00:13:36,000
So begin with the four easy
solutions, and then fill in the

189
00:13:38,000 --> 00:13:44,000
rest.
Now, which are the easy

190
00:13:42,000 --> 00:13:48,000
solutions?
The easy solutions are c1

191
00:13:47,000 --> 00:13:53,000
equals plus or minus 1,
c2 equals zero,

192
00:13:53,000 --> 00:13:59,000
or c1 equals zero,
or c1 = 0, c2 equals plus or

193
00:14:00,000 --> 00:14:06,000
minus 1.
By choosing those four values

194
00:14:05,000 --> 00:14:11,000
of c1 and c2,
I get simple solutions

195
00:14:07,000 --> 00:14:13,000
corresponding to the normal
mode.

196
00:14:10,000 --> 00:14:16,000
If c1 is one and c2 is zero,
I am talking about (1,

197
00:14:14,000 --> 00:14:20,000
negative 1) e to the minus 3t,

198
00:14:17,000 --> 00:14:23,000
and that is very easy plot.
Let's start plotting them.

199
00:14:21,000 --> 00:14:27,000
What I am going to do is
color-code them so you will be

200
00:14:25,000 --> 00:14:31,000
able to recognize what it is I
am plotting.

201
00:14:30,000 --> 00:14:36,000
Let's see.
What colors should we use?

202
00:14:33,000 --> 00:14:39,000
We will use pink and orange.
This will be our pink solution

203
00:14:39,000 --> 00:14:45,000
and our orange solution will be
this one.

204
00:14:43,000 --> 00:14:49,000
Let's plot the pink solution
first.

205
00:14:47,000 --> 00:14:53,000
The pink solution corresponds
to c1 equals 1 and c2

206
00:14:53,000 --> 00:14:59,000
equals zero.
Now, that solution looks like--

207
00:14:58,000 --> 00:15:04,000
Let's write it in pink.

208
00:15:01,000 --> 00:15:07,000
No, let's not write it in pink.
What is the solution?

209
00:15:06,000 --> 00:15:12,000
It looks like x equals e to the
negative 3t,

210
00:15:11,000 --> 00:15:17,000
y equals minus e to the minus
3t.

211
00:15:15,000 --> 00:15:21,000
Well, that's not a good way to
look at it, actually.

212
00:15:19,000 --> 00:15:25,000
The best way to look at it is
to say at t equals zero,

213
00:15:24,000 --> 00:15:30,000
where is it?
It is at the point 1,

214
00:15:26,000 --> 00:15:32,000
negative 1.

215
00:15:30,000 --> 00:15:36,000
And what is it doing as t
increases?

216
00:15:32,000 --> 00:15:38,000
Well, it keeps the direction,
but travels.

217
00:15:35,000 --> 00:15:41,000
The amplitude,
the distance from the origin

218
00:15:39,000 --> 00:15:45,000
keeps shrinking.
As t increases,

219
00:15:41,000 --> 00:15:47,000
this factor,
so it is the tip of this

220
00:15:44,000 --> 00:15:50,000
vector, except the vector is
shrinking.

221
00:15:47,000 --> 00:15:53,000
It is still in the direction of
1, negative 1,

222
00:15:51,000 --> 00:15:57,000
but it is shrinking in
length because its amplitude is

223
00:15:55,000 --> 00:16:01,000
shrinking according to the law e
to the negative 3t.

224
00:16:02,000 --> 00:16:08,000
In other words,
this curve looks like this.

225
00:16:05,000 --> 00:16:11,000
At t equals zero it is over
here, and it goes along this

226
00:16:09,000 --> 00:16:15,000
diagonal line until as t equals
infinity, it gets to infinity,

227
00:16:14,000 --> 00:16:20,000
it reaches the origin.
Of course, it never gets there.

228
00:16:18,000 --> 00:16:24,000
It goes slower and slower and
slower in order that it may

229
00:16:23,000 --> 00:16:29,000
never reach the origin.
What was it doing for values of

230
00:16:27,000 --> 00:16:33,000
t less than zero?
The same thing,

231
00:16:31,000 --> 00:16:37,000
except it was further away.
It comes in from infinity along

232
00:16:35,000 --> 00:16:41,000
that straight line.
In other words,

233
00:16:37,000 --> 00:16:43,000
the eigenvector determines the
line on which it travels and the

234
00:16:41,000 --> 00:16:47,000
eigenvalue determines which way
it goes.

235
00:16:44,000 --> 00:16:50,000
If the eigenvalue is negative,
it is approaching the origin as

236
00:16:48,000 --> 00:16:54,000
t increases.
How about the other one?

237
00:16:51,000 --> 00:16:57,000
Well, if c1 is negative 1,
then everything is the

238
00:16:55,000 --> 00:17:01,000
same except it is the mirror
image of this one.

239
00:17:00,000 --> 00:17:06,000
If c1 is negative 1,
then at t equals zero it is at

240
00:17:03,000 --> 00:17:09,000
this point.
And, once again,

241
00:17:05,000 --> 00:17:11,000
the same reasoning shows that
it is coming into the origin as

242
00:17:10,000 --> 00:17:16,000
t increases.
I have now two solutions,

243
00:17:12,000 --> 00:17:18,000
this one corresponding to c1
equals 1,

244
00:17:16,000 --> 00:17:22,000
and the other one c2 equals
zero.

245
00:17:19,000 --> 00:17:25,000
This one corresponds to c1
equals negative 1.

246
00:17:22,000 --> 00:17:28,000
How about the other guy,
the orange guy?

247
00:17:25,000 --> 00:17:31,000
Well, now c1 is zero,
c2 is one, let's say.

248
00:17:30,000 --> 00:17:36,000
It is the vector (1,
0), but otherwise everything is

249
00:17:33,000 --> 00:17:39,000
the same.
I start now at the point (1,

250
00:17:36,000 --> 00:17:42,000
0) at time zero.
And, as t increases,

251
00:17:39,000 --> 00:17:45,000
I come into the origin always
along that direction.

252
00:17:42,000 --> 00:17:48,000
And before that I came in from
infinity.

253
00:17:45,000 --> 00:17:51,000
And, again, if c2 is 1
and if c2 is negative 1,

254
00:17:50,000 --> 00:17:56,000
I do the same thing but
on the other side.

255
00:18:00,000 --> 00:18:06,000
That wasn't very hard.
I plotted four solutions.

256
00:18:04,000 --> 00:18:10,000
And now I roll up my sleeves
and waive my hands to try to get

257
00:18:10,000 --> 00:18:16,000
others.
The general philosophy is the

258
00:18:14,000 --> 00:18:20,000
following.
The general philosophy is the

259
00:18:18,000 --> 00:18:24,000
differential equation looks like
this.

260
00:18:21,000 --> 00:18:27,000
It is a system of differential
equations.

261
00:18:25,000 --> 00:18:31,000
These are continuous functions.
That means when I draw the

262
00:18:31,000 --> 00:18:37,000
velocity field corresponding to
that system of differential

263
00:18:36,000 --> 00:18:42,000
equations, because their
functions are continuous,

264
00:18:39,000 --> 00:18:45,000
as I move from one (x,
y) point to another the

265
00:18:43,000 --> 00:18:49,000
direction of the velocity
vectors change continuously.

266
00:18:46,000 --> 00:18:52,000
It never suddenly reverses
without something like that.

267
00:18:50,000 --> 00:18:56,000
Now, if that changes
continuously then the

268
00:18:53,000 --> 00:18:59,000
trajectories must change
continuously,

269
00:18:56,000 --> 00:19:02,000
too.
In other words,

270
00:18:59,000 --> 00:19:05,000
nearby trajectories should be
doing approximately the same

271
00:19:03,000 --> 00:19:09,000
thing.
Well, that means all the other

272
00:19:05,000 --> 00:19:11,000
trajectories are ones which come
like that must be going also

273
00:19:10,000 --> 00:19:16,000
toward the origin.
If I start here,

274
00:19:12,000 --> 00:19:18,000
probably I have to follow this
one.

275
00:19:15,000 --> 00:19:21,000
They are all coming to the
origin, but that is a little too

276
00:19:19,000 --> 00:19:25,000
vague.
How do they come to the origin?

277
00:19:22,000 --> 00:19:28,000
In other words,
are they coming in straight

278
00:19:25,000 --> 00:19:31,000
like that?
Probably not.

279
00:19:26,000 --> 00:19:32,000
Then what are they doing?
Now we are coming to the only

280
00:19:32,000 --> 00:19:38,000
point in the lecture which you
might find a little difficult.

281
00:19:36,000 --> 00:19:42,000
Try to follow what I am doing
now.

282
00:19:38,000 --> 00:19:44,000
If you don't follow,
it is not well done in the

283
00:19:42,000 --> 00:19:48,000
textbook, but it is very well
done in the notes because I

284
00:19:46,000 --> 00:19:52,000
wrote them myself.
Please, it is done very

285
00:19:49,000 --> 00:19:55,000
carefully in the notes,
patiently follow through the

286
00:19:52,000 --> 00:19:58,000
explanation.
It takes about that much space.

287
00:19:55,000 --> 00:20:01,000
It is one of the important
ideas that your engineering

288
00:19:59,000 --> 00:20:05,000
professors will expect you to
understand.

289
00:20:04,000 --> 00:20:10,000
Anyway, I know this only from
the negative one because they

290
00:20:08,000 --> 00:20:14,000
say to me at lunch,
ruin my lunch by saying I said

291
00:20:12,000 --> 00:20:18,000
it to my students and got
nothing but blank looks.

292
00:20:16,000 --> 00:20:22,000
What do you guys teach them
over there?

293
00:20:19,000 --> 00:20:25,000
Blah, blah, blah.
Maybe we ought to start

294
00:20:22,000 --> 00:20:28,000
teaching it ourselves.
Sure.

295
00:20:25,000 --> 00:20:31,000
Why don't they start cutting
their own hair,

296
00:20:28,000 --> 00:20:34,000
too?

297
00:20:35,000 --> 00:20:41,000
Here is the idea.
Let me recopy that solution.

298
00:20:40,000 --> 00:20:46,000
The solution looks like (1,
negative 1) e to the minus 3t

299
00:20:46,000 --> 00:20:52,000
plus c2, (1, 0) e to the
negative t.

300
00:20:56,000 --> 00:21:02,000
What I ask is as t goes to
infinity, I feel sure that the

301
00:21:00,000 --> 00:21:06,000
trajectories must be coming into
the origin because these guys

302
00:21:04,000 --> 00:21:10,000
are doing that.
And, in fact,

303
00:21:06,000 --> 00:21:12,000
that is confirmed.
As t goes to infinity,

304
00:21:09,000 --> 00:21:15,000
this goes to zero and that goes
to zero regardless of what the

305
00:21:13,000 --> 00:21:19,000
c1 and c2 are.
That makes it clear that this

306
00:21:17,000 --> 00:21:23,000
goes to zero no matter what the
c1 and c2 are as t goes to

307
00:21:21,000 --> 00:21:27,000
infinity, but I would like to
analyze it a little more

308
00:21:25,000 --> 00:21:31,000
carefully.
As t goes to infinity,

309
00:21:28,000 --> 00:21:34,000
I have the sum of two terms.
And what I ask is,

310
00:21:32,000 --> 00:21:38,000
which term is dominant?
Of these two terms,

311
00:21:36,000 --> 00:21:42,000
are they of equal importance,
or is one more important than

312
00:21:41,000 --> 00:21:47,000
the other?
When t is 10,

313
00:21:43,000 --> 00:21:49,000
for example,
that is not very far on the way

314
00:21:47,000 --> 00:21:53,000
to infinity, but it is certainly
far enough to illustrate.

315
00:21:52,000 --> 00:21:58,000
Well, e to the minus 10
is an extremely

316
00:21:56,000 --> 00:22:02,000
small number.
The only thing smaller is e to

317
00:22:01,000 --> 00:22:07,000
the minus 30.
The term that dominates,

318
00:22:05,000 --> 00:22:11,000
they are both small,
but relatively-speaking this

319
00:22:08,000 --> 00:22:14,000
one is much larger because this
one only has the factor e to the

320
00:22:13,000 --> 00:22:19,000
minus 10,
whereas, this has the factor e

321
00:22:17,000 --> 00:22:23,000
to the minus 30,
which is vanishingly small.

322
00:22:22,000 --> 00:22:28,000
In other words,
as t goes to infinity --

323
00:22:26,000 --> 00:22:32,000
Well, let's write it the other
way.

324
00:22:28,000 --> 00:22:34,000
This is the dominant term,
as t goes to infinity.

325
00:22:38,000 --> 00:22:44,000
Now, just the opposite is true
as t goes to minus infinity.

326
00:22:43,000 --> 00:22:49,000
t going to minus infinity means
I am backing up along these

327
00:22:48,000 --> 00:22:54,000
curves.
As t goes to minus infinity,

328
00:22:51,000 --> 00:22:57,000
let's say t gets to be negative
100, this is e to the 100,

329
00:22:56,000 --> 00:23:02,000
but this is e to the 300,

330
00:23:01,000 --> 00:23:07,000
which is much,
much bigger.

331
00:23:03,000 --> 00:23:09,000
So this is the dominant term as
t goes to negative infinity.

332
00:23:18,000 --> 00:23:24,000
Now what I have is the sum of
two vectors.

333
00:23:20,000 --> 00:23:26,000
Let's first look at what
happens as t goes to infinity.

334
00:23:24,000 --> 00:23:30,000
As t goes to infinity,
I have the sum of two vectors.

335
00:23:28,000 --> 00:23:34,000
This one is completely
negligible compared with the one

336
00:23:31,000 --> 00:23:37,000
on the right-hand side.
In other words,

337
00:23:35,000 --> 00:23:41,000
for a all intents and purposes,
as t goes to infinity,

338
00:23:38,000 --> 00:23:44,000
it is this thing that takes
over.

339
00:23:41,000 --> 00:23:47,000
Therefore, what does the
solution look like as t goes to

340
00:23:45,000 --> 00:23:51,000
infinity?
The answer is it follows the

341
00:23:47,000 --> 00:23:53,000
yellow line.
Now, what does it look like as

342
00:23:50,000 --> 00:23:56,000
it backs up?
As it came in from negative

343
00:23:53,000 --> 00:23:59,000
infinity, what does it look
like?

344
00:23:56,000 --> 00:24:02,000
Now, this one is a little
harder to see.

345
00:24:00,000 --> 00:24:06,000
This is big,
but this is infinity bigger.

346
00:24:03,000 --> 00:24:09,000
I mean very,
very much bigger,

347
00:24:06,000 --> 00:24:12,000
when t is a large negative
number.

348
00:24:09,000 --> 00:24:15,000
Therefore, what I have is the
sum of a very big vector.

349
00:24:14,000 --> 00:24:20,000
You're standing on the moon
looking at the blackboard,

350
00:24:19,000 --> 00:24:25,000
so this is really big.
This is a very big vector.

351
00:24:24,000 --> 00:24:30,000
This is one million meters
long, and this is only 20

352
00:24:29,000 --> 00:24:35,000
meters long.
That is this guy,

353
00:24:33,000 --> 00:24:39,000
and that is this guy.
I want the sum of those two.

354
00:24:36,000 --> 00:24:42,000
What does the sum look like?
The answer is a sum is

355
00:24:40,000 --> 00:24:46,000
approximately parallel to the
long guy because this is

356
00:24:44,000 --> 00:24:50,000
negligible.
This does not mean they are

357
00:24:47,000 --> 00:24:53,000
next to each other.
They are slightly tilted over,

358
00:24:51,000 --> 00:24:57,000
but not very much.
In other words,

359
00:24:53,000 --> 00:24:59,000
as t goes to negative infinity
it doesn't coincide with this

360
00:24:58,000 --> 00:25:04,000
vector.
The solution doesn't,

361
00:25:01,000 --> 00:25:07,000
but it is parallel to it.
It has the same direction.

362
00:25:05,000 --> 00:25:11,000
I am done.
It means far away from the

363
00:25:07,000 --> 00:25:13,000
origin, it should be parallel to
the pink line.

364
00:25:11,000 --> 00:25:17,000
Near the origin it should turn
and become more or less

365
00:25:15,000 --> 00:25:21,000
coincident with the orange line.
And those were the solutions.

366
00:25:19,000 --> 00:25:25,000
That's how they look.

367
00:25:27,000 --> 00:25:33,000
How about down here?
The same thing,

368
00:25:30,000 --> 00:25:36,000
like that, but then after a
while they turn and join.

369
00:25:35,000 --> 00:25:41,000
Here, they have to turn around
to join up, but they join.

370
00:25:41,000 --> 00:25:47,000
And that is,
in a simple way,

371
00:25:44,000 --> 00:25:50,000
the sketches of those
functions.

372
00:25:47,000 --> 00:25:53,000
That is how they must look.
What does this say about our

373
00:25:53,000 --> 00:25:59,000
state?
Well, it says that the fact

374
00:25:57,000 --> 00:26:03,000
that the governor of New
Hampshire is indifferent to what

375
00:26:01,000 --> 00:26:07,000
Massachusetts is doing produces
ultimately harmony.

376
00:26:06,000 --> 00:26:12,000
Both states revert ultimately
their normal advertising budgets

377
00:26:11,000 --> 00:26:17,000
in spite of the fact that
Massachusetts is keeping an eye

378
00:26:15,000 --> 00:26:21,000
peeled out for the slightest
misbehavior on the part of New

379
00:26:20,000 --> 00:26:26,000
Hampshire.
Peace reins,

380
00:26:22,000 --> 00:26:28,000
in other words.
Now you should know some names.

381
00:26:27,000 --> 00:26:33,000
Let's see.
I will write names in purple.

382
00:26:30,000 --> 00:26:36,000
There are two words that are
used to describe this situation.

383
00:26:35,000 --> 00:26:41,000
First is the word that
describes the general pattern of

384
00:26:40,000 --> 00:26:46,000
the way these lines look.
The word for that is a node.

385
00:26:44,000 --> 00:26:50,000
And the fact that all the
trajectories end up at the

386
00:26:48,000 --> 00:26:54,000
origin for that one uses the
word sink.

387
00:26:52,000 --> 00:26:58,000
This could be modified to nodal
sink.

388
00:26:55,000 --> 00:27:01,000
That would be better.
Nodal sink, let's say.

389
00:27:00,000 --> 00:27:06,000
Nodal sink or,
if you like to write them in

390
00:27:03,000 --> 00:27:09,000
the opposite order,
sink node.

391
00:27:06,000 --> 00:27:12,000
In the same way there would be
something called a source node

392
00:27:11,000 --> 00:27:17,000
if I reversed all the arrows.
I am not going to calculate an

393
00:27:16,000 --> 00:27:22,000
example.
Why don't I simply do it by

394
00:27:19,000 --> 00:27:25,000
giving you --
For example,

395
00:27:23,000 --> 00:27:29,000
if the matrix A produced a
solution instead of that one.

396
00:27:28,000 --> 00:27:34,000
Suppose it looked like 1,
negative 1 e to the 3t.

397
00:27:32,000 --> 00:27:38,000
The eigenvalues were reversed,

398
00:27:36,000 --> 00:27:42,000
were now positive.
And I will make the other one

399
00:27:41,000 --> 00:27:47,000
positive, too.
c2 1, 0 e to the t.

400
00:27:57,000 --> 00:28:03,000
What would that change in the
picture?

401
00:27:59,000 --> 00:28:05,000
The answer is essentially
nothing, except the direction of

402
00:28:04,000 --> 00:28:10,000
the arrows.
In other words,

403
00:28:06,000 --> 00:28:12,000
the first thing would still be
1, negative 1.

404
00:28:09,000 --> 00:28:15,000
The only difference is that now

405
00:28:12,000 --> 00:28:18,000
as t increases we go the other
way.

406
00:28:15,000 --> 00:28:21,000
And here the same thing,
we have still the same basic

407
00:28:19,000 --> 00:28:25,000
vector, the same basic orange
vector, orange line,

408
00:28:22,000 --> 00:28:28,000
but it has now traversed the
solution.

409
00:28:25,000 --> 00:28:31,000
We traverse it in the opposite
direction.

410
00:28:30,000 --> 00:28:36,000
Now, let's do the same thing
about dominance,

411
00:28:35,000 --> 00:28:41,000
as we did before.
Which term dominates as t goes

412
00:28:40,000 --> 00:28:46,000
to infinity?
This is the dominant term.

413
00:28:44,000 --> 00:28:50,000
Because, as t goes to infinity,
3t is much bigger than t.

414
00:28:51,000 --> 00:28:57,000
This one, on the other hand,
dominates as t goes to negative

415
00:28:57,000 --> 00:29:03,000
infinity.

416
00:29:05,000 --> 00:29:11,000
How now will the solutions look
like?

417
00:29:07,000 --> 00:29:13,000
Well, as t goes to infinity,
they follow the pink curve.

418
00:29:11,000 --> 00:29:17,000
Whereas, as t starts out from
negative infinity,

419
00:29:15,000 --> 00:29:21,000
they follow the orange curve.

420
00:29:28,000 --> 00:29:34,000
As t goes to infinity,
they become parallel to the

421
00:29:33,000 --> 00:29:39,000
pink curve, and as t goes to
negative infinity,

422
00:29:38,000 --> 00:29:44,000
they are very close to the
origin and are following the

423
00:29:44,000 --> 00:29:50,000
yellow curve.
This is pink and this is

424
00:29:48,000 --> 00:29:54,000
yellow.
They look like this.

425
00:30:03,000 --> 00:30:09,000
Notice the picture basically is
the same.

426
00:30:06,000 --> 00:30:12,000
It is the picture of a node.
All that has happened is the

427
00:30:11,000 --> 00:30:17,000
arrows are reversed.
And, therefore,

428
00:30:14,000 --> 00:30:20,000
this would be called a nodal
source.

429
00:30:17,000 --> 00:30:23,000
The word source and sink
correspond to what you learned

430
00:30:21,000 --> 00:30:27,000
in 18.02 and 8.02,
I hope, also,

431
00:30:24,000 --> 00:30:30,000
or you could call it a source
node.

432
00:30:27,000 --> 00:30:33,000
Both phrases are used,
depending on how you want to

433
00:30:31,000 --> 00:30:37,000
use it in a sentence.
And another word for this,

434
00:30:37,000 --> 00:30:43,000
this would be called unstable
because all of the solutions

435
00:30:41,000 --> 00:30:47,000
starting out from near the
origin ultimately end up

436
00:30:45,000 --> 00:30:51,000
infinitely far away from the
origin.

437
00:30:47,000 --> 00:30:53,000
This would be called stable.
In fact, it would be called

438
00:30:52,000 --> 00:30:58,000
asymptotically stable.
I don't like the word

439
00:30:55,000 --> 00:31:01,000
asymptotically,
but it has become standard in

440
00:30:58,000 --> 00:31:04,000
the literature.
And, more important,

441
00:31:02,000 --> 00:31:08,000
it is standard in your
textbook.

442
00:31:05,000 --> 00:31:11,000
And I don't like to fight with
a textbook.

443
00:31:08,000 --> 00:31:14,000
It just ends up confusing
everybody, including me.

444
00:31:12,000 --> 00:31:18,000
That is enough for nodes.
I would like to talk now about

445
00:31:16,000 --> 00:31:22,000
some of the other cases that can
occur because they lead to

446
00:31:21,000 --> 00:31:27,000
completely different pictures
that you should understand.

447
00:31:26,000 --> 00:31:32,000
Let's look at the case where
our governors behave a little

448
00:31:30,000 --> 00:31:36,000
more badly, a little more
combatively.

449
00:31:40,000 --> 00:31:46,000
It is x prime equals negative x
as before,

450
00:31:46,000 --> 00:31:52,000
but this time a firm response
by Massachusetts to any sign of

451
00:31:52,000 --> 00:31:58,000
increased activity by
stockpiling of advertising

452
00:31:58,000 --> 00:32:04,000
budgets.
Here let's say New Hampshire

453
00:32:03,000 --> 00:32:09,000
now is even worse.
Five times, quintuple or

454
00:32:08,000 --> 00:32:14,000
whatever increase Massachusetts
makes, of course they don't have

455
00:32:15,000 --> 00:32:21,000
an income tax,
but they will manage.

456
00:32:19,000 --> 00:32:25,000
Minus 3y as before.
Let's again calculate quickly

457
00:32:24,000 --> 00:32:30,000
what the characteristic equation
is.

458
00:32:30,000 --> 00:32:36,000
Our matrix is now negative 1,
3, 5 and negative 3.

459
00:32:34,000 --> 00:32:40,000
The characteristic equation now

460
00:32:37,000 --> 00:32:43,000
is lambda squared.
What is that?

461
00:32:40,000 --> 00:32:46,000
Again, plus 4 lambda.
But now the determinant is 3

462
00:32:44,000 --> 00:32:50,000
minus 15 is negative 12.

463
00:32:48,000 --> 00:32:54,000
And this, because I prepared
very carefully,

464
00:32:52,000 --> 00:32:58,000
all eigenvalues are integers.
And so this factors into lambda

465
00:32:57,000 --> 00:33:03,000
plus 6 times lambda minus 2,

466
00:33:01,000 --> 00:33:07,000
does it not?
Yes.

467
00:33:04,000 --> 00:33:10,000
6 lambda minus 2 is four
lambda.

468
00:33:07,000 --> 00:33:13,000
Good.
What do we have?

469
00:33:10,000 --> 00:33:16,000
Well, first of all we have our
eigenvalue lambda,

470
00:33:15,000 --> 00:33:21,000
negative 6.
And the eigenvector that goes

471
00:33:19,000 --> 00:33:25,000
with that is minus 1.
This is negative 1 minus

472
00:33:24,000 --> 00:33:30,000
negative 6 which makes,
shut your eyes,

473
00:33:28,000 --> 00:33:34,000
5.
We have 5a1 plus 3a2 is zero.

474
00:33:32,000 --> 00:33:38,000
And the other equation,

475
00:33:35,000 --> 00:33:41,000
I hope it comes out to be
something similar.

476
00:33:38,000 --> 00:33:44,000
I didn't check.
I am hoping this is right.

477
00:33:42,000 --> 00:33:48,000
The eigenvector is,
okay, you have been taught to

478
00:33:46,000 --> 00:33:52,000
always make one of the 1,
forget about that.

479
00:33:49,000 --> 00:33:55,000
Just pick numbers that make it
come out right.

480
00:33:53,000 --> 00:33:59,000
I am going to make this one 3,
and then I will make this one

481
00:33:57,000 --> 00:34:03,000
negative 5.
As I say, I have a policy of

482
00:34:02,000 --> 00:34:08,000
integers only.
I am a number theorist at

483
00:34:06,000 --> 00:34:12,000
heart.
That is how I started out life

484
00:34:09,000 --> 00:34:15,000
anyway.
There we have data from which

485
00:34:12,000 --> 00:34:18,000
we can make one solution.
How about the other one?

486
00:34:17,000 --> 00:34:23,000
The other one will correspond
to the eigenvalue lambda equals

487
00:34:23,000 --> 00:34:29,000
2.
This time the equation is

488
00:34:25,000 --> 00:34:31,000
negative 1 minus 2 is negative
3.

489
00:34:30,000 --> 00:34:36,000
It is minus 3a1 plus 3a2 is
zero.

490
00:34:34,000 --> 00:34:40,000
And now the eigenvector is (1,
1).

491
00:34:37,000 --> 00:34:43,000
Now we are ready to draw
pictures.

492
00:34:40,000 --> 00:34:46,000
We are going to make this
similar analysis,

493
00:34:44,000 --> 00:34:50,000
but it will go faster now
because you have already had the

494
00:34:49,000 --> 00:34:55,000
experience of that.
First of all,

495
00:34:52,000 --> 00:34:58,000
what is our general solution?
It is going to be c1 times 3,

496
00:34:57,000 --> 00:35:03,000
negative 5 e to the minus 6t.

497
00:35:02,000 --> 00:35:08,000
And then the other normal mode

498
00:35:06,000 --> 00:35:12,000
times an arbitrary constant will
be 1, 1 times e to the 2t.

499
00:35:18,000 --> 00:35:24,000
I am going to use the same
strategy.

500
00:35:20,000 --> 00:35:26,000
We have our two normal modes
here, eigenvalue,

501
00:35:24,000 --> 00:35:30,000
eigenvector solutions from
which, by adjusting these

502
00:35:27,000 --> 00:35:33,000
constants, we can get our four
basic solutions.

503
00:35:32,000 --> 00:35:38,000
Those are going to look like,
let's draw a picture here.

504
00:35:37,000 --> 00:35:43,000
Again, I will color-code them.
Let's use pink again.

505
00:35:42,000 --> 00:35:48,000
The pink solution now starts at
3, negative 5.

506
00:35:47,000 --> 00:35:53,000
That is where it is when t is

507
00:35:50,000 --> 00:35:56,000
zero.
And, because of the coefficient

508
00:35:54,000 --> 00:36:00,000
minus 6 up there,
it is coming into the origin

509
00:35:58,000 --> 00:36:04,000
and looks like that.
And its mirror image,

510
00:36:03,000 --> 00:36:09,000
of course, does the same thing.
That is when c1 is negative

511
00:36:08,000 --> 00:36:14,000
one.
How about the orange guy?

512
00:36:10,000 --> 00:36:16,000
Well, when t is equal to zero,
it is at 1, 1.

513
00:36:14,000 --> 00:36:20,000
But what is it doing after

514
00:36:16,000 --> 00:36:22,000
that?
As t increases,

515
00:36:18,000 --> 00:36:24,000
it is getting further away from
the origin because the sign here

516
00:36:22,000 --> 00:36:28,000
is positive.
e to the 2t is

517
00:36:25,000 --> 00:36:31,000
increasing, it is not decreasing
anymore, so this guy is going

518
00:36:30,000 --> 00:36:36,000
out.
And its mirror image on the

519
00:36:35,000 --> 00:36:41,000
other side is doing the same
thing.

520
00:36:40,000 --> 00:36:46,000
Now all we have to do is fill
in the picture.

521
00:36:46,000 --> 00:36:52,000
Well, you fill it in by
continuity.

522
00:36:51,000 --> 00:36:57,000
Your nearby trajectories must
be doing what similar thing?

523
00:37:00,000 --> 00:37:06,000
If I start out very near the
pink guy, I should stay near the

524
00:37:04,000 --> 00:37:10,000
pink guy.
But as I get near the origin,

525
00:37:07,000 --> 00:37:13,000
I am also approaching the
orange guy.

526
00:37:09,000 --> 00:37:15,000
Well, there is no other
possibility other than that.

527
00:37:13,000 --> 00:37:19,000
If you are further away you
start turning a little sooner.

528
00:37:17,000 --> 00:37:23,000
I am just using an argument
from continuity to say the

529
00:37:21,000 --> 00:37:27,000
picture must be roughly filled
out this way.

530
00:37:24,000 --> 00:37:30,000
Maybe not exactly.
In fact, there are fine points.

531
00:37:29,000 --> 00:37:35,000
And I am going to ask you to do
one of them on Friday for the

532
00:37:32,000 --> 00:37:38,000
new problem set,
even before the exam,

533
00:37:35,000 --> 00:37:41,000
God forbid.
But I want you to get a little

534
00:37:37,000 --> 00:37:43,000
more experience working with
that linear phase portrait

535
00:37:41,000 --> 00:37:47,000
visual because it is,
I think, one of the best ones

536
00:37:44,000 --> 00:37:50,000
this semester.
You can learn a lot from it.

537
00:37:47,000 --> 00:37:53,000
Anyway, you are not done with
it, but I hope you have at least

538
00:37:51,000 --> 00:37:57,000
looked at it by now.
That is what the picture looks

539
00:37:54,000 --> 00:38:00,000
like.
First of all,

540
00:37:55,000 --> 00:38:01,000
what are we going to name this?
In other words,

541
00:38:00,000 --> 00:38:06,000
forget about the arrows.
If you just look at the general

542
00:38:05,000 --> 00:38:11,000
way those lines go,
where have you seen this

543
00:38:08,000 --> 00:38:14,000
before?
You saw this in 18.02.

544
00:38:11,000 --> 00:38:17,000
What was the topic?
You were plotting contour

545
00:38:15,000 --> 00:38:21,000
curves of functions,
were you not?

546
00:38:18,000 --> 00:38:24,000
What did you call contours
curves that formed that pattern?

547
00:38:23,000 --> 00:38:29,000
A saddle point.
You called this a saddle point

548
00:38:26,000 --> 00:38:32,000
because it was like the center
of a saddle.

549
00:38:32,000 --> 00:38:38,000
It is like a mountain pass.
Here you are going up the

550
00:38:35,000 --> 00:38:41,000
mountain, say,
and here you are going down,

551
00:38:37,000 --> 00:38:43,000
the way the contour line is
going down.

552
00:38:40,000 --> 00:38:46,000
And this is sort of a min and
max point.

553
00:38:42,000 --> 00:38:48,000
A maximum if you go in that
direction and a minimum if you

554
00:38:46,000 --> 00:38:52,000
go in that direction,
say.

555
00:38:48,000 --> 00:38:54,000
Without the arrows on it,
it is like a saddle point.

556
00:38:51,000 --> 00:38:57,000
And so the same word is used
here.

557
00:38:53,000 --> 00:38:59,000
It is called the saddle.
You don't say point in the same

558
00:38:56,000 --> 00:39:02,000
way you don't say a nodal point.
It is the whole picture,

559
00:39:01,000 --> 00:39:07,000
as it were, that is the saddle.
It is a saddle.

560
00:39:05,000 --> 00:39:11,000
There is the saddle.
This is where you sit.

561
00:39:08,000 --> 00:39:14,000
Now, should I call it a source
or a sink?

562
00:39:12,000 --> 00:39:18,000
I cannot call it either because
it is a sink along these lines,

563
00:39:16,000 --> 00:39:22,000
it is a source along those
lines and along the others,

564
00:39:21,000 --> 00:39:27,000
it starts out looking like a
sink and then turns around and

565
00:39:25,000 --> 00:39:31,000
starts acting like a source.
The word source and sink are

566
00:39:31,000 --> 00:39:37,000
not used for saddle.
The only word that is used is

567
00:39:34,000 --> 00:39:40,000
unstable because definitely it
is unstable.

568
00:39:38,000 --> 00:39:44,000
If you start off exactly on the
pink lines you do end up at the

569
00:39:42,000 --> 00:39:48,000
origin, but if you start
anywhere else ever so close to a

570
00:39:47,000 --> 00:39:53,000
pink line you think you are
going to the origin,

571
00:39:50,000 --> 00:39:56,000
but then at the last minute you
are zooming off out to infinity

572
00:39:55,000 --> 00:40:01,000
again.
This is a typical example of

573
00:39:57,000 --> 00:40:03,000
instability.
Only if you do the

574
00:40:01,000 --> 00:40:07,000
mathematically possible,
but physically impossible thing

575
00:40:06,000 --> 00:40:12,000
of starting out exactly on the
pink line, only then will you

576
00:40:11,000 --> 00:40:17,000
get to the origin.
If you start out anywhere else,

577
00:40:15,000 --> 00:40:21,000
make the slightest error in
measure and get off the pink

578
00:40:20,000 --> 00:40:26,000
line, you end off at infinity.
What is the effect with our

579
00:40:25,000 --> 00:40:31,000
war-like governors fighting for
the tourist trade willing to

580
00:40:30,000 --> 00:40:36,000
spend any amounts of money to
match and overmatch what their

581
00:40:35,000 --> 00:40:41,000
competitor in the nearby state
is spending?

582
00:40:41,000 --> 00:40:47,000
The answer is,
they all lose.

583
00:40:43,000 --> 00:40:49,000
Since it is mostly this section
of the diagram that makes sense,

584
00:40:48,000 --> 00:40:54,000
what happens is they end up all
spending an infinity of dollars

585
00:40:53,000 --> 00:40:59,000
and nobody gets any more
tourists than anybody else.

586
00:40:58,000 --> 00:41:04,000
So this is a model of what not
to do.

587
00:41:02,000 --> 00:41:08,000
I have one more model to show
you.

588
00:41:05,000 --> 00:41:11,000
Maybe we better start over at
this board here.

589
00:41:11,000 --> 00:41:17,000
Massachusetts on top.
New Hampshire on the bottom.

590
00:41:17,000 --> 00:41:23,000
x prime is going to be,
that is Massachusetts,

591
00:41:23,000 --> 00:41:29,000
I guess as before.
Let me get the numbers right.

592
00:41:45,000 --> 00:41:51,000
Leave that out for a moment.
y prime is 2x minus 3y.

593
00:41:50,000 --> 00:41:56,000
New Hampshire behaves normally.

594
00:41:54,000 --> 00:42:00,000
It is ready to respond to
anything Massachusetts can put

595
00:41:59,000 --> 00:42:05,000
out.
But by itself,

596
00:42:01,000 --> 00:42:07,000
it really wants to bring its
budget to normal.

597
00:42:05,000 --> 00:42:11,000
Now, Massachusetts,
we have a Mormon governor now,

598
00:42:09,000 --> 00:42:15,000
I guess.
Imagine instead we have a

599
00:42:12,000 --> 00:42:18,000
Buddhist governor.
A Buddhist governor reacts as

600
00:42:16,000 --> 00:42:22,000
follows, minus y.
What does that mean?

601
00:42:20,000 --> 00:42:26,000
It means that when he sees New
Hampshire increasing the budget,

602
00:42:25,000 --> 00:42:31,000
his reaction is,
we will lower ours.

603
00:42:30,000 --> 00:42:36,000
We will show them love.
It looks suicidal,

604
00:42:34,000 --> 00:42:40,000
but what actually happens?
Well, our little program is

605
00:42:39,000 --> 00:42:45,000
over.
Our matrix a is negative 1,

606
00:42:42,000 --> 00:42:48,000
negative 1, 2,
negative 3.

607
00:42:46,000 --> 00:42:52,000
The characteristic equations is

608
00:42:50,000 --> 00:42:56,000
lambda squared plus 4 lambda.

609
00:42:55,000 --> 00:43:01,000
And now what is the other term?
3 minus negative 2 makes 5.

610
00:43:02,000 --> 00:43:08,000
This is not going to factor
because I tried it out and I

611
00:43:07,000 --> 00:43:13,000
know it is not going to factor.
We are going to get lambda

612
00:43:13,000 --> 00:43:19,000
equals, we will just use the
quadratic formula,

613
00:43:17,000 --> 00:43:23,000
negative 4 plus or minus the
square root of 16 minus 4 times

614
00:43:23,000 --> 00:43:29,000
5, that is 16 minus 20 or
negative 4 all divided by 2,

615
00:43:28,000 --> 00:43:34,000
which makes minus 2,
pull out the 4,

616
00:43:31,000 --> 00:43:37,000
that makes it a 2,
cancels this 2,

617
00:43:35,000 --> 00:43:41,000
minus 1 inside.
It is minus 2 plus or minus i.

618
00:43:40,000 --> 00:43:46,000
Complex solutions.

619
00:43:44,000 --> 00:43:50,000
What are we doing to do about
that?

620
00:43:47,000 --> 00:43:53,000
Well, you should rejoice when
you get this case and are asked

621
00:43:53,000 --> 00:43:59,000
to sketch it because,
even if you calculate the

622
00:43:58,000 --> 00:44:04,000
complex eigenvector and from
that take its real and imaginary

623
00:44:04,000 --> 00:44:10,000
parts of the complex solution,
in fact, you will not be able

624
00:44:10,000 --> 00:44:16,000
easily to sketch the answer
anyway.

625
00:44:15,000 --> 00:44:21,000
But let me show you what sort
of thing you can get and then I

626
00:44:18,000 --> 00:44:24,000
am going to wave my hands and
argue a little bit to try to

627
00:44:21,000 --> 00:44:27,000
indicate what it is that the
solution actually looks like.

628
00:44:24,000 --> 00:44:30,000
You are going to get something
that looks like --

629
00:44:28,000 --> 00:44:34,000
A typical real solution is
going to look like this.

630
00:44:31,000 --> 00:44:37,000
This is going to produce e to
the minus 2t times e

631
00:44:36,000 --> 00:44:42,000
to the i t.
e to the minus 2 plus i all

632
00:44:40,000 --> 00:44:46,000
times t.
This will be our exponential

633
00:44:44,000 --> 00:44:50,000
factor which is shrinking in
amplitude.

634
00:44:47,000 --> 00:44:53,000
This is going to give me sines
and cosines.

635
00:44:50,000 --> 00:44:56,000
When I separate out the
eigenvector into its real and

636
00:44:54,000 --> 00:45:00,000
imaginary parts,
it is going to look something

637
00:44:57,000 --> 00:45:03,000
like this.
a1, a2 times cosine t,

638
00:45:02,000 --> 00:45:08,000
that is from the e to the it

639
00:45:05,000 --> 00:45:11,000
part.
Then there will be a sine term.

640
00:45:08,000 --> 00:45:14,000
And all that is going to be
multiplied by the exponential

641
00:45:12,000 --> 00:45:18,000
factor e to the negative 2t.

642
00:45:22,000 --> 00:45:28,000
That is just one normal mode.
It is going to be c1 times this

643
00:45:28,000 --> 00:45:34,000
plus c2 times something similar.
It doesn't matter exactly what

644
00:45:34,000 --> 00:45:40,000
it is because they are all going
to look the same.

645
00:45:37,000 --> 00:45:43,000
Namely, this is a shrinking
amplitude.

646
00:45:40,000 --> 00:45:46,000
I am not going to worry about
that.

647
00:45:42,000 --> 00:45:48,000
My real question is,
what does this look like?

648
00:45:45,000 --> 00:45:51,000
In other words,
as a pair of parametric

649
00:45:48,000 --> 00:45:54,000
equations, if x is equal to a1
cosine t plus b1 sine t

650
00:45:52,000 --> 00:45:58,000
and y is a2
cosine plus b2 sine,

651
00:45:56,000 --> 00:46:02,000
what does it look like?

652
00:46:01,000 --> 00:46:07,000
Well, what are its
characteristics?

653
00:46:03,000 --> 00:46:09,000
In the first place,
as a curve this part of it is

654
00:46:08,000 --> 00:46:14,000
bounded.
It stays within some large box

655
00:46:11,000 --> 00:46:17,000
because cosine and sine never
get bigger than one and never

656
00:46:16,000 --> 00:46:22,000
get smaller than minus one.
It is periodic.

657
00:46:20,000 --> 00:46:26,000
As t increases to t plus 2pi,

658
00:46:24,000 --> 00:46:30,000
it comes back to exactly the
same point it was at before.

659
00:46:35,000 --> 00:46:41,000
We have a curve that is
repeating itself periodically,

660
00:46:38,000 --> 00:46:44,000
it does not go off to infinity.
And here is where I am waving

661
00:46:42,000 --> 00:46:48,000
my hands.
It satisfies an equation.

662
00:46:44,000 --> 00:46:50,000
Those of you who like to fool
around with mathematics a little

663
00:46:49,000 --> 00:46:55,000
bit, it is not difficult to show
this, but it satisfies an

664
00:46:52,000 --> 00:46:58,000
equation of the form A x squared
plus B y squared plus C xy

665
00:46:56,000 --> 00:47:02,000
equals D.

666
00:47:00,000 --> 00:47:06,000
All you have to do is figure
out what the coefficients A,

667
00:47:03,000 --> 00:47:09,000
B, C and D should be.
And the way to do it is,

668
00:47:06,000 --> 00:47:12,000
if you calculate the square of
x you are going to get cosine

669
00:47:10,000 --> 00:47:16,000
squared, sine squared and a
cosine sine term.

670
00:47:13,000 --> 00:47:19,000
You are going to get those same
three terms here and the same

671
00:47:17,000 --> 00:47:23,000
three terms here.
You just use undetermined

672
00:47:20,000 --> 00:47:26,000
coefficients,
set up a system of simultaneous

673
00:47:23,000 --> 00:47:29,000
equations and you will be able
to find the A,

674
00:47:26,000 --> 00:47:32,000
B, C and D that work.
I am looking for a curve that

675
00:47:31,000 --> 00:47:37,000
is bounded, keeps repeating its
values and that satisfies a

676
00:47:35,000 --> 00:47:41,000
quadratic equation which looks
like this.

677
00:47:38,000 --> 00:47:44,000
Well, an earlier generation
would know from high school,

678
00:47:42,000 --> 00:47:48,000
these curves are all conic
sections.

679
00:47:45,000 --> 00:47:51,000
The only curves that satisfy
equations like that are

680
00:47:48,000 --> 00:47:54,000
hyperbola, parabolas,
the conic sections in other

681
00:47:52,000 --> 00:47:58,000
words, and ellipses.
Circles are a special kind of

682
00:47:56,000 --> 00:48:02,000
ellipses.
There is a degenerate case.

683
00:48:00,000 --> 00:48:06,000
A pair of lines which can be
considered a degenerate

684
00:48:04,000 --> 00:48:10,000
hyperbola, if you want.
It is as much a hyperbola as a

685
00:48:08,000 --> 00:48:14,000
circle, as an ellipse say.
Which of these is it?

686
00:48:11,000 --> 00:48:17,000
Well, it must be those guys.
Those are the only guys that

687
00:48:16,000 --> 00:48:22,000
stay bounded and repeat
themselves periodically.

688
00:48:20,000 --> 00:48:26,000
The other guys don't do that.
These are ellipses.

689
00:48:23,000 --> 00:48:29,000
And, therefore,
what do they look like?

690
00:48:28,000 --> 00:48:34,000
Well, they must look like an
ellipse that is trying to be an

691
00:48:32,000 --> 00:48:38,000
ellipse, but each time it goes
around the point is pulled a

692
00:48:37,000 --> 00:48:43,000
little closer to the origin.
It must be doing this,

693
00:48:41,000 --> 00:48:47,000
in other words.
And such a point is called a

694
00:48:44,000 --> 00:48:50,000
spiral sink.
Again sink because,

695
00:48:47,000 --> 00:48:53,000
no matter where you start,
you will get a curve that

696
00:48:51,000 --> 00:48:57,000
spirals into the origin.
Spiral is self-explanatory.

697
00:48:55,000 --> 00:49:01,000
And the one thing I haven't
told you that you must read is

698
00:49:00,000 --> 00:49:06,000
how do you know that it goes
around counterclockwise and not

699
00:49:04,000 --> 00:49:10,000
clockwise?
Read clockwise or

700
00:49:08,000 --> 00:49:14,000
counterclockwise.
I will give you the answer in

701
00:49:12,000 --> 00:49:18,000
30 seconds, not for this
particular curve.

702
00:49:16,000 --> 00:49:22,000
That you will have to
calculate.

703
00:49:19,000 --> 00:49:25,000
All you have to do is put in
somewhere.

704
00:49:23,000 --> 00:49:29,000
Let's say at the point (1,
0), a single vector from the

705
00:49:28,000 --> 00:49:34,000
velocity field.
In other words,

706
00:49:32,000 --> 00:49:38,000
at the point (1,
0), when x is 1 and y is 0 our

707
00:49:37,000 --> 00:49:43,000
vector is minus 1, 2,

708
00:49:40,000 --> 00:49:46,000
which is the vector minus 1,
2, it goes like this.

709
00:49:46,000 --> 00:49:52,000
Therefore, the motion must be
counterclockwise.

710
00:49:51,000 --> 00:49:57,000
And, by the way,
what is the effect of having a

711
00:49:56,000 --> 00:50:02,000
Buddhist governor?
Peace.

712
00:50:00,000 --> 00:50:06,000
Everything spirals into the
origin and everybody is left

713
00:50:05,000 --> 00:50:11,000
with the same advertising budget
they always had.

714
00:50:10,000 --> 00:50:16,000
Thanks.