1
00:00:00 --> 00:00:06

2
00:00:08 --> 00:00:14
First of all,
the way a nonlinear autonomous

3
00:00:11 --> 00:00:17
system looks,
you have had some practice with

4
00:00:15 --> 00:00:21
it by now.
This is nonlinear.

5
00:00:17 --> 00:00:23
The right-hand side are no
longer simple combinations ax

6
00:00:22 --> 00:00:28
plus by.
Nonlinear and autonomous,

7
00:00:26 --> 00:00:32
these are function just of x
and y.

8
00:00:30 --> 00:00:36
There is no t on the right-hand
side.

9
00:00:33 --> 00:00:39
Now, most of today will be
geometric.

10
00:00:37 --> 00:00:43
The way to get a geometric
picture of that is first by

11
00:00:43 --> 00:00:49
constructing the velocity field
whose components are the

12
00:00:48 --> 00:00:54
functions f and g.
This is a velocity field that

13
00:00:53 --> 00:00:59
gives a picture of the system
and has solutions.

14
00:01:00 --> 00:01:06
The solutions to the system,
from the point of view of

15
00:01:05 --> 00:01:11
functions, they would look like
pairs of functions,

16
00:01:09 --> 00:01:15
x of t, y of t.

17
00:01:12 --> 00:01:18
But, from the point of view of
geometry, when you plot them as

18
00:01:18 --> 00:01:24
parametric equations,
they are called trajectories of

19
00:01:23 --> 00:01:29
the field F, which simply means
that they are curves everywhere

20
00:01:29 --> 00:01:35
having the right velocity.
So a typical curve would look

21
00:01:34 --> 00:01:40
like --
There is a trajectory.

22
00:01:38 --> 00:01:44
And we know it is a trajectory
because at each point the vector

23
00:01:42 --> 00:01:48
on it has, of course,
the right direction,

24
00:01:45 --> 00:01:51
the tangent direction,
but more than that,

25
00:01:48 --> 00:01:54
it has the right velocity.
So here, for example,

26
00:01:51 --> 00:01:57
the point is traveling more
slowly.

27
00:01:53 --> 00:01:59
Here it is traveling more
rapidly because the velocity

28
00:01:57 --> 00:02:03
vector is bigger,
longer.

29
00:02:00 --> 00:02:06
So this is a picture of a
typical trajectory.

30
00:02:03 --> 00:02:09
The only other things that I
should mention are the critical

31
00:02:09 --> 00:02:15
points.
If you have worked the problems

32
00:02:12 --> 00:02:18
for this week,
the first couple of problems,

33
00:02:16 --> 00:02:22
you have already seen the
significance of the critical

34
00:02:21 --> 00:02:27
points.
Well, from Monday's lecture you

35
00:02:24 --> 00:02:30
know from the point of view of
solutions they are constant

36
00:02:29 --> 00:02:35
solutions.

37
00:02:32 --> 00:02:38

38
00:02:37 --> 00:02:43
From the point of view of the
field they are where the field

39
00:02:41 --> 00:02:47
is zero.
There is no velocity vector,

40
00:02:43 --> 00:02:49
in other words.
The velocity vector is zero.

41
00:02:46 --> 00:02:52
And, therefore,
a point being there has no

42
00:02:49 --> 00:02:55
reason to go anywhere else.
And, spelling it out,

43
00:02:53 --> 00:02:59
it's where the partial
derivatives, where the values of

44
00:02:57 --> 00:03:03
the functions on the right-hand
side, which give the two

45
00:03:01 --> 00:03:07
components, the i and j
components of the field,

46
00:03:04 --> 00:03:10
where they are zero.

47
00:03:07 --> 00:03:13

48
00:03:11 --> 00:03:17
That is all I will need by way
of a recall today.

49
00:03:14 --> 00:03:20
I don't think I will need
anything else.

50
00:03:17 --> 00:03:23
The topic for today is another
kind of behavior that you have

51
00:03:22 --> 00:03:28
not yet observed at the computer
screen, unless you have worked

52
00:03:26 --> 00:03:32
ahead, and that is there are
trajectories which go along to

53
00:03:31 --> 00:03:37
infinity or end up at a critical
point.

54
00:03:35 --> 00:03:41
They are the critical points
that just sit there all the

55
00:03:39 --> 00:03:45
time.
But there is a third type of

56
00:03:42 --> 00:03:48
behavior that a trajectory can
have where it neither sits for

57
00:03:47 --> 00:03:53
all time nor goes off for all
time.

58
00:03:50 --> 00:03:56
Instead, it repeats itself.
Such a thing is called a closed

59
00:03:55 --> 00:04:01
trajectory.
What does it look like?

60
00:03:59 --> 00:04:05
Well, it is a closed curve in
the plane that at every point,

61
00:04:05 --> 00:04:11
it is a trajectory,
i.e., the arrows at each point,

62
00:04:10 --> 00:04:16
let's say it is traced in the
clockwise direction.

63
00:04:15 --> 00:04:21
And so the arrows of the field
will go like this.

64
00:04:20 --> 00:04:26
Here it is going slowly,
here it is very slow and here

65
00:04:25 --> 00:04:31
it picks up a little speed again
and so on.

66
00:04:31 --> 00:04:37
Now, for such a trajectory what
is happening?

67
00:04:35 --> 00:04:41
Well, it goes around in finite
time and then repeats itself.

68
00:04:42 --> 00:04:48
It just goes round and round
forever if you land on that

69
00:04:48 --> 00:04:54
trajectory.
It represents a system that

70
00:04:53 --> 00:04:59
returns to its original state
periodically.

71
00:04:57 --> 00:05:03
It represents periodic behavior
of the system.

72
00:05:04 --> 00:05:10

73
00:05:16 --> 00:05:22
Now, we have seen one example
of that, a simple example where

74
00:05:23 --> 00:05:29
this simple system,
x prime equals y,

75
00:05:29 --> 00:05:35
y prime equals negative x.

76
00:05:35 --> 00:05:41

77
00:05:40 --> 00:05:46
We could write down the
solutions to that directly,

78
00:05:43 --> 00:05:49
but if you want to do
eigenvalues and eigenvectors the

79
00:05:47 --> 00:05:53
matrix will look like this.
The equation will be lambda

80
00:05:51 --> 00:05:57
squared plus zero lambda plus
one equals zero,

81
00:05:55 --> 00:06:01
so the eigenvalues will be plus

82
00:05:58 --> 00:06:04
or minus i.
In fact, from then on you could

83
00:06:02 --> 00:06:08
work out in the usual ways the
eigenvectors,

84
00:06:04 --> 00:06:10
complex eigenvectors and
separate them.

85
00:06:07 --> 00:06:13
But, look, you can avoid all
that just by writing down the

86
00:06:11 --> 00:06:17
solution.
The solutions are sines and

87
00:06:13 --> 00:06:19
cosines.
One basic solution will be x

88
00:06:16 --> 00:06:22
equals cosine t,
in which case what is y?

89
00:06:19 --> 00:06:25
Well, y is the derivative of
that.

90
00:06:22 --> 00:06:28
That will be minus sine t.

91
00:06:24 --> 00:06:30
Another basic solution,
we will start with x equals

92
00:06:28 --> 00:06:34
sine t.
In which case y will be cosine

93
00:06:33 --> 00:06:39
t, its derivative.

94
00:06:36 --> 00:06:42
Now, if you do that,
what do these things look like?

95
00:06:41 --> 00:06:47
Well, either of these two basic
solutions looks like a circle,

96
00:06:47 --> 00:06:53
not traced in the usual way but
in the opposite way.

97
00:06:52 --> 00:06:58
For example,
when t is equal to zero it

98
00:06:56 --> 00:07:02
starts at the point one,
zero.

99
00:07:00 --> 00:07:06
Now, if the minus sign were not
there this would be x equals

100
00:07:04 --> 00:07:10
cosine t,
y equals sine t,

101
00:07:08 --> 00:07:14
which is the usual
counterclockwise circle.

102
00:07:11 --> 00:07:17
But if I change y from sine t
to negative sine t

103
00:07:15 --> 00:07:21
it is going around the
other way.

104
00:07:18 --> 00:07:24
So this circle is traced that
way.

105
00:07:20 --> 00:07:26
And this is a family of
circles, according to the values

106
00:07:24 --> 00:07:30
of c1 and c2,
concentric, all of which go

107
00:07:27 --> 00:07:33
around clockwise.
So those are closed

108
00:07:31 --> 00:07:37
trajectories.
Those are the solutions.

109
00:07:33 --> 00:07:39
They are trajectories of the
vector field.

110
00:07:36 --> 00:07:42
They are closed.
They come around and they

111
00:07:39 --> 00:07:45
repeat in finite time.
Now, these are no good.

112
00:07:42 --> 00:07:48
These are the kind I am not
interested in.

113
00:07:45 --> 00:07:51
These are commonplace,
and we are interested in good

114
00:07:49 --> 00:07:55
stuff today.
And the good stuff we are

115
00:07:51 --> 00:07:57
interested in is limit cycles.

116
00:07:55 --> 00:08:01

117
00:08:01 --> 00:08:07
A limit cycle is a closed
trajectory with a couple of

118
00:08:06 --> 00:08:12
extra hypotheses.
It is a closed trajectory,

119
00:08:10 --> 00:08:16
just like those guys,
but it has something they don't

120
00:08:15 --> 00:08:21
have, namely,
it is king of the roost.

121
00:08:19 --> 00:08:25
They have to be isolated,
no other guys nearby.

122
00:08:23 --> 00:08:29
And they also have to be
stable.

123
00:08:27 --> 00:08:33
See, the problem here is that
none of these stands out from

124
00:08:32 --> 00:08:38
any of the others.
In other words,

125
00:08:37 --> 00:08:43
there must be,
isolated means,

126
00:08:40 --> 00:08:46
no others nearby.

127
00:08:42 --> 00:08:48

128
00:08:49 --> 00:08:55
That is just what goes wrong
here.

129
00:08:51 --> 00:08:57
Arbitrarily close to each of
these circles is yet another

130
00:08:55 --> 00:09:01
circle doing exactly the same
thing.

131
00:08:58 --> 00:09:04
That means that there are some
that are only of mild interest.

132
00:09:03 --> 00:09:09
What is much more interesting
is to find a cycle where there

133
00:09:07 --> 00:09:13
is nothing nearby.
Something, therefore,

134
00:09:10 --> 00:09:16
that looks like this.

135
00:09:13 --> 00:09:19

136
00:09:22 --> 00:09:28
Here is our pink guy.
Let's make this one go

137
00:09:25 --> 00:09:31
counterclockwise.
Here is a limit cycle,

138
00:09:28 --> 00:09:34
it seems to be.
And now what do nearby guys do?

139
00:09:33 --> 00:09:39
Well, they should approach it.
Somebody here like that does

140
00:09:38 --> 00:09:44
this, spirals in and gets ever
and every closer to that thing.

141
00:09:44 --> 00:09:50
Now, it can never join it
because, if it joined it at the

142
00:09:50 --> 00:09:56
joining point,
I would have two solutions

143
00:09:54 --> 00:10:00
going through this point.
And that is illegal.

144
00:10:00 --> 00:10:06
All it can do is get
arbitrarily close.

145
00:10:02 --> 00:10:08
On the computer screen it will
look as if it joins it but,

146
00:10:06 --> 00:10:12
of course, it cannot.
It is just the resolution,

147
00:10:10 --> 00:10:16
the pixels.
Not enough pixels.

148
00:10:12 --> 00:10:18
The resolution isn't good
enough.

149
00:10:14 --> 00:10:20
And the ones that start further
away will take longer to find

150
00:10:18 --> 00:10:24
their way to the limit cycle and
they will always stay outside of

151
00:10:23 --> 00:10:29
the earlier guys,
but they will get arbitrarily

152
00:10:26 --> 00:10:32
close, too.
How about inside?

153
00:10:30 --> 00:10:36
Inside, well,
it starts somewhere and does

154
00:10:33 --> 00:10:39
the same thing.
It starts here and will try to

155
00:10:37 --> 00:10:43
join the limit cycle.
That is what I mean by

156
00:10:41 --> 00:10:47
stability.
Stability means that nearby

157
00:10:44 --> 00:10:50
guys, the guys that start
somewhere else eventually

158
00:10:48 --> 00:10:54
approach the limit cycle,
regardless of whether they

159
00:10:52 --> 00:10:58
start from the outside or start
from the inside.

160
00:10:56 --> 00:11:02
So that is stable.
An unstable limit cycle --

161
00:11:02 --> 00:11:08
But I am not calling it a limit
cycle if it is unstable.

162
00:11:06 --> 00:11:12
I am just calling it a closed
trajectory, but let's draw one

163
00:11:10 --> 00:11:16
which is unstable.
Here is the way we will look if

164
00:11:14 --> 00:11:20
it is unstable.
Guys that start nearby will be

165
00:11:17 --> 00:11:23
repelled, driven somewhere else.
Or, if they start here,

166
00:11:21 --> 00:11:27
they will go away from the
thing instead of going toward

167
00:11:25 --> 00:11:31
it.
This is unstable.

168
00:11:27 --> 00:11:33
And I don't call it a limit
cycle.

169
00:11:29 --> 00:11:35
It is just a closed trajectory.

170
00:11:33 --> 00:11:39

171
00:11:38 --> 00:11:44
Cycle because it cycles round
and round.

172
00:11:40 --> 00:11:46
Limit because it is the limit
of the nearby curves.

173
00:11:44 --> 00:11:50
The other case where it is
unstable is not the limit.

174
00:11:48 --> 00:11:54
Of course, you could have a
case also where the curves

175
00:11:52 --> 00:11:58
outside spiral in toward it but
the ones inside are repelled and

176
00:11:57 --> 00:12:03
do this.
That would be called

177
00:11:59 --> 00:12:05
semi-stable.
And you can make up all sorts

178
00:12:03 --> 00:12:09
of cases.
And I think I,

179
00:12:05 --> 00:12:11
at one point,
drew them in the notes,

180
00:12:07 --> 00:12:13
but I am not going to.
The only interesting one,

181
00:12:11 --> 00:12:17
of permanent importance that
people study,

182
00:12:14 --> 00:12:20
are the actual limit cycles.
No, it was the stable closed

183
00:12:19 --> 00:12:25
trajectories.
Notice, by the way,

184
00:12:21 --> 00:12:27
a closed trajectory is always a
simple curve.

185
00:12:25 --> 00:12:31
Remember what that means from
18.02?

186
00:12:29 --> 00:12:35
Simple means it doesn't cross
itself.

187
00:12:32 --> 00:12:38
Why doesn't it cross itself?
It cannot cross itself because,

188
00:12:37 --> 00:12:43
if it tried to,
what is wrong with that point?

189
00:12:41 --> 00:12:47
At that point which way does
the vector field go,

190
00:12:46 --> 00:12:52
that way or that way?
Why the interest of limit

191
00:12:50 --> 00:12:56
cycles?
Well, because there are systems

192
00:12:54 --> 00:13:00
in nature in which just this
type of behavior,

193
00:12:58 --> 00:13:04
they have a certain periodic
motion.

194
00:13:03 --> 00:13:09
And, if you disturb it,
gradually it returns to its

195
00:13:07 --> 00:13:13
original periodic state.
A simple example is breathing.

196
00:13:11 --> 00:13:17
Now I have made you all
self-conscious.

197
00:13:14 --> 00:13:20
All of you are breathing.
If you are here you are

198
00:13:18 --> 00:13:24
breathing.
At what rate are you breathing?

199
00:13:22 --> 00:13:28
Well, you are unaware of it,
of course, except now.

200
00:13:26 --> 00:13:32
If you are sitting here
listening.

201
00:13:30 --> 00:13:36
There is a certain temperature
and a certain air circulation in

202
00:13:34 --> 00:13:40
the room.
You are not thinking of

203
00:13:36 --> 00:13:42
anything, certainly not of the
lecture, and the lecture is not

204
00:13:40 --> 00:13:46
unduly exciting,
you will breathe at a certain

205
00:13:43 --> 00:13:49
steady rate which is a little
different for every person but

206
00:13:47 --> 00:13:53
that is your rate.
Now, you can artificially

207
00:13:50 --> 00:13:56
change that.
You could say now I am going to

208
00:13:53 --> 00:13:59
breathe faster.
And indeed you can.

209
00:13:57 --> 00:14:03
But, as soon as you stop being
aware of what you are doing,

210
00:14:01 --> 00:14:07
the levels of various hormones
and carbon dioxide in your

211
00:14:06 --> 00:14:12
bloodstream and so on will
return your breathing to its

212
00:14:11 --> 00:14:17
natural state.
In other words,

213
00:14:13 --> 00:14:19
that system of your breathing,
which is controlled by various

214
00:14:18 --> 00:14:24
chemicals and hormones in the
body, is exhibiting exactly this

215
00:14:23 --> 00:14:29
type of behavior.
It has a certain regular

216
00:14:27 --> 00:14:33
periodic motion as a system.
And, if disturbed,

217
00:14:32 --> 00:14:38
if artificially you set it out
somewhere else,

218
00:14:34 --> 00:14:40
it will gradually return to its
original state.

219
00:14:37 --> 00:14:43
Now, of course,
if I am running it will be

220
00:14:40 --> 00:14:46
different.
Sure.

221
00:14:41 --> 00:14:47
If you are running you breathe
faster, but that is because the

222
00:14:45 --> 00:14:51
parameters in the system,
the a's and the b's in the

223
00:14:48 --> 00:14:54
equation, the f of (x,
y) and g of (x,

224
00:14:50 --> 00:14:56
y), the parameters in those

225
00:14:53 --> 00:14:59
functions will be set at
different levels.

226
00:14:56 --> 00:15:02
You will have different
hormones, a different of carbon

227
00:14:59 --> 00:15:05
dioxide and so on.
Now, I am not saying that

228
00:15:04 --> 00:15:10
breathing is modeled by a limit
cycle.

229
00:15:07 --> 00:15:13
It is the sort of thing which
one might look for a limit

230
00:15:11 --> 00:15:17
cycle.
That is, of course,

231
00:15:13 --> 00:15:19
a question for biologists.
And, in general,

232
00:15:17 --> 00:15:23
any type of periodic behavior
in nature, people try to see if

233
00:15:22 --> 00:15:28
there is some system of
differential equations which

234
00:15:26 --> 00:15:32
governs it in which perhaps
there is a limit cycle,

235
00:15:30 --> 00:15:36
which contains a limit cycle.
Well, what are the problems?

236
00:15:36 --> 00:15:42
In a sense, limit cycles are
easy to lecture about because so

237
00:15:41 --> 00:15:47
little is known about them.
At the end of the period,

238
00:15:46 --> 00:15:52
if I have time,
I will show you that the

239
00:15:49 --> 00:15:55
simplest possible question you
could ask, the answer to it is

240
00:15:55 --> 00:16:01
totally known after 120 years of
steady trying.

241
00:16:00 --> 00:16:06
But let's first talk about what
sorts of problems people address

242
00:16:05 --> 00:16:11
with limit cycles.
First of all is the existence

243
00:16:09 --> 00:16:15
problem.

244
00:16:10 --> 00:16:16

245
00:16:16 --> 00:16:22
If I give you a system,
you know, the right-hand side

246
00:16:19 --> 00:16:25
is x squared plus 2y cubed minus
3xy,

247
00:16:23 --> 00:16:29
and the g is something similar.
I say does this have limit

248
00:16:26 --> 00:16:32
cycles?
Well, you know how to find its

249
00:16:29 --> 00:16:35
critical points.
But how do you find out if it

250
00:16:34 --> 00:16:40
has limit cycles?
The answer to that is nobody

251
00:16:40 --> 00:16:46
has any idea.
This problem,

252
00:16:44 --> 00:16:50
in general, there are not much
in the way of methods.

253
00:16:50 --> 00:16:56
Not much.

254
00:16:53 --> 00:16:59

255
00:17:00 --> 00:17:06
Not much is known.
There is one theorem that you

256
00:17:03 --> 00:17:09
will find in the notes,
a simple theorem called the

257
00:17:07 --> 00:17:13
Poincare-Bendixson theorem
which, for about 60 or 70 years

258
00:17:11 --> 00:17:17
was about the only result known
which enabled people to find

259
00:17:16 --> 00:17:22
limit cycles.
Nowadays the theorem is used

260
00:17:19 --> 00:17:25
relatively little because people
try to find limit cycles by

261
00:17:24 --> 00:17:30
computer.
Now, the difficulty is you have

262
00:17:27 --> 00:17:33
to know where to look for them.
In other words,

263
00:17:32 --> 00:17:38
the computer screen shows that
much and you set the axes and it

264
00:17:36 --> 00:17:42
doesn't show any limit cycles.
That doesn't mean there are not

265
00:17:40 --> 00:17:46
any.
That means they are over there,

266
00:17:43 --> 00:17:49
or it means there is a big one
like there.

267
00:17:46 --> 00:17:52
And you are looking in the
middle of it and don't see it.

268
00:17:50 --> 00:17:56
So, in general,
people don't look for limit

269
00:17:53 --> 00:17:59
cycles unless the physical
system that gave rise to the

270
00:17:56 --> 00:18:02
pair of differential equations
suggests that there is something

271
00:18:01 --> 00:18:07
repetitive going on like
breathing.

272
00:18:05 --> 00:18:11
And, if it tells you that,
then it often gives you

273
00:18:09 --> 00:18:15
approximate values of the
parameters and the variables so

274
00:18:14 --> 00:18:20
you know where to look.
Basically this is done by

275
00:18:18 --> 00:18:24
computer search guided by the
physical problem.

276
00:18:23 --> 00:18:29

277
00:18:31 --> 00:18:37
Therefore, I cannot say much
more about it today.

278
00:18:35 --> 00:18:41
Instead I am going to focus my
attention on nonexistence.

279
00:18:40 --> 00:18:46
When can you be sure that a
system will not have any limit

280
00:18:46 --> 00:18:52
cycles?
And there are two theorems.

281
00:18:49 --> 00:18:55
One, again, due to Bendixson
who was a Swedish mathematician

282
00:18:54 --> 00:19:00
who lived around 1900 or so.
There is a criterion due to

283
00:18:59 --> 00:19:05
Bendixson.
And there is one involving

284
00:19:04 --> 00:19:10
critical points.
And I would like to describe

285
00:19:08 --> 00:19:14
both of them for you today.
First of all,

286
00:19:11 --> 00:19:17
Bendixson's criterion.

287
00:19:14 --> 00:19:20

288
00:19:22 --> 00:19:28
It is very simply stated and
has a marvelous proof,

289
00:19:25 --> 00:19:31
which I am going to give you.
We have D as a region of the

290
00:19:29 --> 00:19:35
plane.

291
00:19:30 --> 00:19:36

292
00:19:35 --> 00:19:41
And what Bendixson's criterion
tells you to do is take your

293
00:19:40 --> 00:19:46
vector field and calculate its
divergence.

294
00:19:43 --> 00:19:49
We are set back in 1802,
and this proof is going to be

295
00:19:48 --> 00:19:54
straight 18.02.
You will enjoy it.

296
00:19:51 --> 00:19:57
Calculate the divergence.
Now, I am talking about the

297
00:19:55 --> 00:20:01
two-dimensional divergence.
Remember that is fx,

298
00:20:00 --> 00:20:06
the partial of f with respect
to x, plus the partial of the g,

299
00:20:05 --> 00:20:11
the j component with respect to
y.

300
00:20:08 --> 00:20:14
And assume that that is a
continuous function.

301
00:20:12 --> 00:20:18
It always will be with us.
Practically all the examples I

302
00:20:16 --> 00:20:22
will give you f and g will be
simple polynomials.

303
00:20:20 --> 00:20:26
They are smooth,
continuous and nice and behave

304
00:20:24 --> 00:20:30
as you want.
And you calculate that and

305
00:20:27 --> 00:20:33
assume --
Suppose, in other words,

306
00:20:32 --> 00:20:38
that the divergence of f,
I need more room.

307
00:20:36 --> 00:20:42
The hypothesis is that the
divergence of f is not zero in

308
00:20:42 --> 00:20:48
that region D.
It is never zero.

309
00:20:45 --> 00:20:51
It is not zero at any point in
that region.

310
00:20:49 --> 00:20:55
The conclusion is that there
are no limit cycles in the

311
00:20:55 --> 00:21:01
region.
If it is not zero in D,

312
00:20:58 --> 00:21:04
there are no limit cycles.
In fact, there are not even any

313
00:21:04 --> 00:21:10
closed trajectories.
You couldn't even have those

314
00:21:09 --> 00:21:15
bunch of concentric circles,
so there are no closed

315
00:21:13 --> 00:21:19
trajectories of the original
system whose divergence you

316
00:21:18 --> 00:21:24
calculated.
There are no closed

317
00:21:21 --> 00:21:27
trajectories in D.
For example,

318
00:21:24 --> 00:21:30
let me give you a simple
example to put a little flesh on

319
00:21:29 --> 00:21:35
it.
Let's see.

320
00:21:32 --> 00:21:38
What do I have?
I prepared an example.

321
00:21:35 --> 00:21:41
x prime equals,
here is a simple nonlinear

322
00:21:40 --> 00:21:46
system, x cubed plus y cubed.

323
00:21:45 --> 00:21:51
And y prime equals 3x plus y
cubed plus 2y.

324
00:21:51 --> 00:21:57

325
00:21:53 --> 00:21:59

326
00:21:59 --> 00:22:05
Does this system have limit
cycles?

327
00:22:01 --> 00:22:07
Well, even to calculate its
critical points would be a

328
00:22:05 --> 00:22:11
little task, but we can easily
answer the question as to

329
00:22:09 --> 00:22:15
whether it has limit cycles or
not by Bendixson's criterion.

330
00:22:14 --> 00:22:20
Let's calculate the divergence.
The divergence of the vector

331
00:22:18 --> 00:22:24
field whose components are these
two functions is,

332
00:22:22 --> 00:22:28
well, 3x squared,
it's the partial of the first

333
00:22:26 --> 00:22:32
guy with respect to x plus the
partial of the second guy with

334
00:22:31 --> 00:22:37
respect to y,
which is 3y squared plus two.

335
00:22:34 --> 00:22:40
Now, can that be zero anywhere

336
00:22:39 --> 00:22:45
in the x,y-plane?
No, because it is the sum of

337
00:22:43 --> 00:22:49
these two squares.
This much of it could be zero

338
00:22:47 --> 00:22:53
only at the origin,
but that plus two eliminates

339
00:22:51 --> 00:22:57
even that.
This is always positive in the

340
00:22:55 --> 00:23:01
entire x,y-plane.
Here my domain is the whole

341
00:22:59 --> 00:23:05
x,y-plane and,
therefore, the conclusion is

342
00:23:03 --> 00:23:09
that there are no closed
trajectories in the x,y-plane,

343
00:23:07 --> 00:23:13
anywhere.

344
00:23:10 --> 00:23:16

345
00:23:15 --> 00:23:21
And we have done that with just
a couple of lines of calculation

346
00:23:18 --> 00:23:24
and nothing further required.
No computer search.

347
00:23:21 --> 00:23:27
In fact, no computer search
could ever proof this.

348
00:23:24 --> 00:23:30
It would be impossible because,
no matter where you look,

349
00:23:27 --> 00:23:33
there is always some other
place to look.

350
00:23:30 --> 00:23:36
This is an example where a
couple lines of mathematics

351
00:23:35 --> 00:23:41
dispose of the matter far more
effectively than a million

352
00:23:41 --> 00:23:47
dollars worth of calculation.
Well, where does Bendixson's

353
00:23:46 --> 00:23:52
theorem come from?
Yes, Bendixson's theorem comes

354
00:23:51 --> 00:23:57
from 18.02.
And I am giving it to you both

355
00:23:56 --> 00:24:02
to recall a little bit of 18.02
to you.

356
00:24:01 --> 00:24:07
Because it is about the first
example in the course that we

357
00:24:06 --> 00:24:12
have had of an indirect
argument.

358
00:24:09 --> 00:24:15
And indirect arguments are
something you have to slowly get

359
00:24:14 --> 00:24:20
used to.
I am going to give you an

360
00:24:17 --> 00:24:23
indirect proof.
Remember what that is?

361
00:24:20 --> 00:24:26
You assume the contrary and you
show it leads to a

362
00:24:25 --> 00:24:31
contradiction.
What would assuming the

363
00:24:28 --> 00:24:34
contrary be?
Contrary would be I will assume

364
00:24:34 --> 00:24:40
the divergence is not zero,
but I will suppose there is a

365
00:24:40 --> 00:24:46
closed trajectory.
Suppose there is a closed

366
00:24:44 --> 00:24:50
trajectory that exists.

367
00:24:48 --> 00:24:54

368
00:24:56 --> 00:25:02
Let's draw a picture of it.

369
00:24:59 --> 00:25:05

370
00:25:08 --> 00:25:14
And let's say it goes around
this way.

371
00:25:11 --> 00:25:17
There is a closed trajectory
for our system.

372
00:25:15 --> 00:25:21
Let's call the curve C.
And I am going to call the

373
00:25:19 --> 00:25:25
inside of it R,
the way one often does in

374
00:25:22 --> 00:25:28
18.02.
D is all this region out here,

375
00:25:25 --> 00:25:31
in which everything is taking
place.

376
00:25:30 --> 00:25:36
This is to exist in D.
Now, what I am going to do is

377
00:25:35 --> 00:25:41
calculate a line integral around
that curve.

378
00:25:40 --> 00:25:46

379
00:25:45 --> 00:25:51
A line integral of this vector
field.

380
00:25:47 --> 00:25:53
Now, there are two things you
can calculate.

381
00:25:51 --> 00:25:57
One of the line integrals,
I will put in a few of the

382
00:25:55 --> 00:26:01
vectors here.
The vectors I know are pointing

383
00:25:58 --> 00:26:04
this way because that is the
direction in which the curve is

384
00:26:03 --> 00:26:09
being traversed in order to make
it a trajectory.

385
00:26:08 --> 00:26:14
Those are a few of the typical
vectors in the field.

386
00:26:12 --> 00:26:18
I am going to calculate the
line integral around that curve

387
00:26:16 --> 00:26:22
in the positive sense.
In other words,

388
00:26:19 --> 00:26:25
not in the direction of the
salmon-colored arrow,

389
00:26:23 --> 00:26:29
but in the normal sense in
which you calculate it using

390
00:26:27 --> 00:26:33
Green's theorem,
for example.

391
00:26:31 --> 00:26:37
The positive sense means the
one which keeps the region,

392
00:26:34 --> 00:26:40
the inside on your left,
as you walk around like that

393
00:26:38 --> 00:26:44
the region stays on your left.
That is the positive sense.

394
00:26:42 --> 00:26:48
That is the sense in which I am
integrating.

395
00:26:45 --> 00:26:51
I am going to use Green's
theorem, but the integral that I

396
00:26:49 --> 00:26:55
am going to calculate is not the
work integral.

397
00:26:52 --> 00:26:58
I am going to calculate instead
the flux integral,

398
00:26:55 --> 00:27:01
the integral that represents
the flux of F across C.

399
00:27:00 --> 00:27:06
Now, what is that integral?
Well, at each point,

400
00:27:04 --> 00:27:10
you station a little ant and
the ant reports the outward flow

401
00:27:09 --> 00:27:15
rate across that point which is
F dotted with the normal vector.

402
00:27:15 --> 00:27:21
I will put in a few normal
vectors just to remind you.

403
00:27:20 --> 00:27:26
The normal vectors look like
little unit vectors pointing

404
00:27:25 --> 00:27:31
perpendicularly outwards
everywhere.

405
00:27:29 --> 00:27:35
These are the n's.
F dotted with the unit normal

406
00:27:34 --> 00:27:40
vector, and that is added up
around the curve.

407
00:27:38 --> 00:27:44
This quantity gives me the flux
of the field across C.

408
00:27:43 --> 00:27:49
Now, we are going to calculate
that by Green's theorem.

409
00:27:48 --> 00:27:54
But, before we calculate it by
Green's theorem,

410
00:27:52 --> 00:27:58
we are going to psych it out.
What is it?

411
00:27:55 --> 00:28:01
What is the value of that
integral?

412
00:28:00 --> 00:28:06
Well, since I am asking you to
do it in your head there can

413
00:28:04 --> 00:28:10
only be one possible answer.
It is zero.

414
00:28:06 --> 00:28:12
Why is that integral zero?
Well, because at each point the

415
00:28:10 --> 00:28:16
field vector,
the velocity vector is

416
00:28:13 --> 00:28:19
perpendicular to the normal
vector.

417
00:28:15 --> 00:28:21
Why?
The normal vector points

418
00:28:17 --> 00:28:23
perpendicularly to the curve but
the field vector always is

419
00:28:22 --> 00:28:28
tangent to the curve because
this curve is a trajectory.

420
00:28:27 --> 00:28:33
It is always supposed to be
going in the direction given by

421
00:28:34 --> 00:28:40
that white field vector.
Do you follow?

422
00:28:39 --> 00:28:45
A trajectory means that it is
always tangent to the field

423
00:28:47 --> 00:28:53
vector and, therefore,
always perpendicular to the

424
00:28:53 --> 00:28:59
normal vector.
This is zero since F dot n is

425
00:28:59 --> 00:29:05
always zero.
Everywhere on the curve,

426
00:29:03 --> 00:29:09
F dot n has to be zero.
There is no flux of this field

427
00:29:07 --> 00:29:13
across the curve because the
field is always in the same

428
00:29:12 --> 00:29:18
direction as the curve,
never perpendicular to it.

429
00:29:15 --> 00:29:21
It has no components
perpendicular to it.

430
00:29:19 --> 00:29:25
Good.
Now let's do it the hard way.

431
00:29:21 --> 00:29:27
Let's use Green's theorem.
Green's theorem says that the

432
00:29:25 --> 00:29:31
flux across C should be equal to
the double integral over that

433
00:29:30 --> 00:29:36
region of the divergence of F.
It's like Gauss theorem in two

434
00:29:35 --> 00:29:41
dimensions, this version of it.
Divergence of F,

435
00:29:39 --> 00:29:45
that is a function,
I double integrate it over the

436
00:29:42 --> 00:29:48
region, and then that is dx /
dy, or let's say da because you

437
00:29:46 --> 00:29:52
might want to do it in polar
coordinates.

438
00:29:48 --> 00:29:54
And, on the problem set,
you certainly will want to do

439
00:29:52 --> 00:29:58
it in polar coordinates,
I think.

440
00:29:55 --> 00:30:01

441
00:30:00 --> 00:30:06
All right.
How much is that?

442
00:30:02 --> 00:30:08
Well, we haven't yet used the
hypothesis.

443
00:30:06 --> 00:30:12
All we have done is set up the
problem.

444
00:30:10 --> 00:30:16
Now, the hypothesis was that
the divergence is never zero

445
00:30:16 --> 00:30:22
anywhere in D.
Therefore, the divergence is

446
00:30:20 --> 00:30:26
never zero anywhere in R.
What I say is the divergence is

447
00:30:26 --> 00:30:32
either greater than zero
everywhere in R.

448
00:30:32 --> 00:30:38
Or less than zero everywhere in
R.

449
00:30:35 --> 00:30:41
But it cannot be sometimes
positive and sometimes negative.

450
00:30:40 --> 00:30:46
Why not?
In other words,

451
00:30:42 --> 00:30:48
I say it is not possible the
divergence here is one and here

452
00:30:47 --> 00:30:53
is minus two.
That is not possible because,

453
00:30:51 --> 00:30:57
if I drew a line from this
point to that,

454
00:30:55 --> 00:31:01
along that line the divergence
would start positive and end up

455
00:31:00 --> 00:31:06
negative.
And, therefore,

456
00:31:04 --> 00:31:10
have to be zero some time in
between.

457
00:31:06 --> 00:31:12
It's because it is a continuous
function.

458
00:31:09 --> 00:31:15
It is a continuous function.
I am assuming that.

459
00:31:12 --> 00:31:18
And, therefore,
if it sometimes positive and

460
00:31:15 --> 00:31:21
sometimes negative it has to be
zero in between.

461
00:31:18 --> 00:31:24
You cannot get continuously
from plus one to minus two

462
00:31:22 --> 00:31:28
without passing through zero.
The reason for this is,

463
00:31:27 --> 00:31:33
since the divergence is never
zero in R it therefore must

464
00:31:35 --> 00:31:41
always stay positive or always
stay negative.

465
00:31:40 --> 00:31:46
Now, if it always stays
positive, the conclusion is then

466
00:31:47 --> 00:31:53
this double integral must be
positive.

467
00:31:52 --> 00:31:58
Therefore, this double integral
is either greater than zero.

468
00:32:01 --> 00:32:07
That is if the divergence is
always positive.

469
00:32:04 --> 00:32:10
Or, it is less than zero if the
divergence is always negative.

470
00:32:10 --> 00:32:16
But the one thing it cannot be
is not zero.

471
00:32:14 --> 00:32:20
Well, the left-hand side,
Green's theorem is supposed to

472
00:32:18 --> 00:32:24
be true.
Green's theorem is our bedrock.

473
00:32:22 --> 00:32:28
18.02 would crumble without
that so it must be true.

474
00:32:26 --> 00:32:32
One way of calculating the
left-hand side gives us zero.

475
00:32:33 --> 00:32:39
If we calculate the right-hand
side it is not zero.

476
00:32:36 --> 00:32:42
That is called the
contradiction.

477
00:32:38 --> 00:32:44
Where did the contradiction
arise from?

478
00:32:41 --> 00:32:47
It arose from the fact that I
supposed that there was a closed

479
00:32:45 --> 00:32:51
trajectory in that region.
The conclusion is there cannot

480
00:32:48 --> 00:32:54
be any closed trajectory of that
region because it leads to a

481
00:32:52 --> 00:32:58
contradiction via Green's
theorem.

482
00:32:56 --> 00:33:02

483
00:33:02 --> 00:33:08
Let me see if I can give you
some of the argument for the

484
00:33:06 --> 00:33:12
other, well, let's at least
state the other criterion I

485
00:33:10 --> 00:33:16
wanted to give you.

486
00:33:12 --> 00:33:18

487
00:33:21 --> 00:33:27
Suppose, for example,
we use this system,

488
00:33:25 --> 00:33:31
x prime equals --

489
00:33:27 --> 00:33:33

490
00:33:50 --> 00:33:56
Does this have limit cycles?

491
00:33:53 --> 00:33:59

492
00:33:59 --> 00:34:05
Does that have limit cycles?

493
00:34:02 --> 00:34:08

494
00:34:08 --> 00:34:14
Let's Bendixson it.
We will calculate the

495
00:34:12 --> 00:34:18
divergence of a vector field.
It is 2x from the top function.

496
00:34:18 --> 00:34:24
The partial with respect to x
is 2x.

497
00:34:22 --> 00:34:28
The second function with
respect to y is negative 2y.

498
00:34:29 --> 00:34:35
That certainly could be zero.
In fact, this is zero along the

499
00:34:34 --> 00:34:40
entire line x equals y.
Its divergence is zero here

500
00:34:39 --> 00:34:45
along that whole line.
The best I could conclude was,

501
00:34:44 --> 00:34:50
I could conclude that there is
no limit cycle like this and

502
00:34:50 --> 00:34:56
there is no limit cycle like
this, but there is nothing so

503
00:34:55 --> 00:35:01
far that says a limit cycle
could not cross that because

504
00:35:00 --> 00:35:06
that would not violate
Bendixson's theorem.

505
00:35:06 --> 00:35:12
In other words,
any domain that contained part

506
00:35:09 --> 00:35:15
of this line,
the divergence would be zero

507
00:35:13 --> 00:35:19
along that line.
And, therefore,

508
00:35:15 --> 00:35:21
I could conclude nothing.
I could have limit cycles that

509
00:35:20 --> 00:35:26
cross that line,
as long as they included a

510
00:35:23 --> 00:35:29
piece of that line in them.
The answer is I cannot make a

511
00:35:28 --> 00:35:34
conclusion.
Well, that is because I am

512
00:35:32 --> 00:35:38
using the wrong criterion.
Let's instead use the critical

513
00:35:36 --> 00:35:42
point criterion.

514
00:35:38 --> 00:35:44

515
00:35:55 --> 00:36:01
Now, I am going to say that it
makes a nice positive statement

516
00:35:58 --> 00:36:04
but nobody ever uses it this
way.

517
00:36:01 --> 00:36:07
Nonetheless,
let's first state it

518
00:36:03 --> 00:36:09
positively, even though that is
not the way to use it.

519
00:36:07 --> 00:36:13
The positive statement will be,
once again, we have our region

520
00:36:13 --> 00:36:19
D and we have a region of the xy
plane and we have our C,

521
00:36:20 --> 00:36:26
a closed trajectory in it.
A closed trajectory of what?

522
00:36:26 --> 00:36:32
Of our system.
And that is supposed to be in

523
00:36:30 --> 00:36:36
D.
The critical point criterion

524
00:36:35 --> 00:36:41
says something very simple.
If you have that situation it

525
00:36:42 --> 00:36:48
says that inside that closed
trajectory there must be a

526
00:36:48 --> 00:36:54
critical point somewhere.

527
00:36:52 --> 00:36:58

528
00:37:00 --> 00:37:06
It says that inside C is a
critical point.

529
00:37:08 --> 00:37:14

530
00:37:15 --> 00:37:21
Now, this won't help us with
the existence problem.

531
00:37:18 --> 00:37:24
This won't help us find a
closed trajectory.

532
00:37:21 --> 00:37:27
We will take our system and say
it has a critical point here and

533
00:37:26 --> 00:37:32
a critical point there.
Does it have a closed

534
00:37:29 --> 00:37:35
trajectory?
Well, all I know is the closed

535
00:37:33 --> 00:37:39
trajectory, if it exists,
will have to go around one or

536
00:37:36 --> 00:37:42
more of those critical points.
But I don't know where.

537
00:37:40 --> 00:37:46
It is not going to go around it
like this.

538
00:37:43 --> 00:37:49
It might go around it like
this.

539
00:37:45 --> 00:37:51
And my computer search won't
find it because it is looking at

540
00:37:49 --> 00:37:55
too small a part of the screen.
It doesn't work that way.

541
00:37:53 --> 00:37:59
It works negatively by
contraposition.

542
00:37:56 --> 00:38:02
Do you know what the
contrapositive is?

543
00:38:00 --> 00:38:06
You will at least learn that.
A implies B says the same thing

544
00:38:07 --> 00:38:13
as not B implies not A.

545
00:38:11 --> 00:38:17

546
00:38:20 --> 00:38:26
They are different statements
but they are equivalent to each

547
00:38:24 --> 00:38:30
other.
If you prove one you prove the

548
00:38:27 --> 00:38:33
other.
What would be the

549
00:38:29 --> 00:38:35
contrapositive here?
If you have a closed trajectory

550
00:38:35 --> 00:38:41
inside is a critical point.
The theorem is used this way.

551
00:38:44 --> 00:38:50
If D has no critical points,
it has no closed trajectories

552
00:38:53 --> 00:38:59
and therefore has no limit
cycle.

553
00:39:00 --> 00:39:06
Because, if it did have a
closed trajectory,

554
00:39:03 --> 00:39:09
inside it would be a critical
point.

555
00:39:07 --> 00:39:13
But I said B had no critical
point.

556
00:39:10 --> 00:39:16
That enables us to dispose of
this system that Bendixson could

557
00:39:15 --> 00:39:21
not handle at all.
We can dispose of this system

558
00:39:20 --> 00:39:26
immediately.
Namely, what is it?

559
00:39:22 --> 00:39:28
Where are its critical points?
Well, where is that zero?

560
00:39:29 --> 00:39:35
x squared plus y 
squared is one,

561
00:39:32 --> 00:39:38
plus one is never zero.
This is positive.

562
00:39:35 --> 00:39:41
Or, worse, zero.
And then I add the one to it

563
00:39:38 --> 00:39:44
and it is not zero anymore.
This has no zeros and,

564
00:39:42 --> 00:39:48
therefore, it does not matter
that this one has a lot of

565
00:39:46 --> 00:39:52
zeros.
It makes no difference.

566
00:39:48 --> 00:39:54
It has no critical points.
It has none,

567
00:39:51 --> 00:39:57
therefore, no limit cycles.

568
00:39:54 --> 00:40:00

569
00:40:01 --> 00:40:07
Now, I desperately wanted to
give you the proof of this.

570
00:40:04 --> 00:40:10
It is clearly impossible in the
time remaining.

571
00:40:07 --> 00:40:13
The proof requires a little
time.

572
00:40:09 --> 00:40:15
I haven't decided what to do
about that.

573
00:40:12 --> 00:40:18
It might leak over until
Friday's lecture.

574
00:40:15 --> 00:40:21
Instead, I will finish by
telling you a story.

575
00:40:18 --> 00:40:24
How is that?

576
00:40:20 --> 00:40:26

577
00:40:37 --> 00:40:43
And along side of it was little
y prime.

578
00:40:39 --> 00:40:45
I am not going to continue on
with the letters of the

579
00:40:43 --> 00:40:49
alphabet.
I will prime the earlier one.

580
00:40:46 --> 00:40:52
This has a total of 12
parameters in it.

581
00:40:49 --> 00:40:55
But, in fact,
if you change variables you can

582
00:40:52 --> 00:40:58
get rid of all the linear terms.
The important part of it is

583
00:40:57 --> 00:41:03
only the quadratic terms in the
beginning.

584
00:41:01 --> 00:41:07
This sort of thing is called a
quadratic system.

585
00:41:05 --> 00:41:11
After you have departed from
linear systems,

586
00:41:09 --> 00:41:15
it is the simplest kind there
is.

587
00:41:12 --> 00:41:18
And the predictor-prey,
the robin-earthworm example I

588
00:41:16 --> 00:41:22
gave you is of a typical
quadratic system and exhibits

589
00:41:21 --> 00:41:27
typical nonlinear quadratic
system behavior.

590
00:41:25 --> 00:41:31
Now, the problem is the
following.

591
00:41:30 --> 00:41:36
A, b, c, d, e,
f and so on,

592
00:41:32 --> 00:41:38
those are just real numbers,
parameters, so I am allowed to

593
00:41:37 --> 00:41:43
give them any values I want.
And the problem that has

594
00:41:42 --> 00:41:48
bothered people since 1880 when
it was first proposed is how

595
00:41:47 --> 00:41:53
many limit cycles can a
quadratic system have?

596
00:41:52 --> 00:41:58

597
00:42:03 --> 00:42:09
After 120 years this problem is
totally unsolved,

598
00:42:07 --> 00:42:13
and the mathematicians of the
world who are interested in it

599
00:42:12 --> 00:42:18
cannot even agree with each
other on what the right

600
00:42:16 --> 00:42:22
conjecture is.
But let me tell you a little

601
00:42:20 --> 00:42:26
bit of its history.
There were attempts to solve it

602
00:42:24 --> 00:42:30
in the 20 or 30 years after it
was first proposed,

603
00:42:28 --> 00:42:34
through the 1920 and `30s which
all seemed to have gaps in them.

604
00:42:35 --> 00:42:41
Until finally around 1950 two
Russians mathematicians,

605
00:42:39 --> 00:42:45
one of whom is extremely
well-known, Petrovski,

606
00:42:44 --> 00:42:50
a specialist in systems of
ordinary differential equations

607
00:42:49 --> 00:42:55
published a long and difficult,
complicated 100 page paper in

608
00:42:54 --> 00:43:00
which they proved that the
maximum number is three.

609
00:43:00 --> 00:43:06
I won't put down their names.
Petrovski-Landis.

610
00:43:03 --> 00:43:09
The maximum number was three.
And then not many people were

611
00:43:08 --> 00:43:14
able to read the paper,
and those who did there seemed

612
00:43:12 --> 00:43:18
to be gaps in the reasoning in
various places until finally

613
00:43:16 --> 00:43:22
Arnold who was the greatest
Russian, in my opinion,

614
00:43:20 --> 00:43:26
one of the greatest Russian
mathematician,

615
00:43:23 --> 00:43:29
certainly in this field of
analysis and differential

616
00:43:27 --> 00:43:33
equations, but in other fields,
too, he still is great,

617
00:43:31 --> 00:43:37
although he is somewhat older
now, criticized it.

618
00:43:37 --> 00:43:43
He said look,
there is a really big gap in

619
00:43:41 --> 00:43:47
this argument and it really
cannot be considered to be

620
00:43:46 --> 00:43:52
proven.
People tried working very hard

621
00:43:50 --> 00:43:56
to patch it up and without
success.

622
00:43:53 --> 00:43:59
Then about 1972 or so,
'75 maybe, a Chinese

623
00:43:58 --> 00:44:04
mathematician found a system
with four.

624
00:44:03 --> 00:44:09
Wrote down the numbers,
the number a is so much,

625
00:44:07 --> 00:44:13
b is so much,
and they were absurd numbers

626
00:44:11 --> 00:44:17
like 10^-6 and 40 billion and so
on, nothing you could plot on a

627
00:44:17 --> 00:44:23
computer screen,
but found a system with four.

628
00:44:22 --> 00:44:28
Nobody after this tried to fill
in the gap in the

629
00:44:26 --> 00:44:32
Petrovski-Landis paper.
I was then chairman of the math

630
00:44:32 --> 00:44:38
department, and one of my tasks
was protocol and so on.

631
00:44:36 --> 00:44:42
Anyway, we were trying very
hard to attract a Chinese

632
00:44:40 --> 00:44:46
mathematician to our department
to become a full professor.

633
00:44:44 --> 00:44:50
He was a really outstanding
analyst and specialist in

634
00:44:48 --> 00:44:54
various fields.
Anyway, he came in for a

635
00:44:50 --> 00:44:56
courtesy interview and we
chatted.

636
00:44:53 --> 00:44:59
At the time,
I was very much interested in

637
00:44:56 --> 00:45:02
limit cycles.
And I had on my desk the Math

638
00:44:59 --> 00:45:05
Society's translation of the
Chinese book on limit cycles.

639
00:45:05 --> 00:45:11
A collection of papers by
Chinese mathematicians all on

640
00:45:08 --> 00:45:14
limit cycles.
After a certain point he said,

641
00:45:11 --> 00:45:17
oh, I see you're interested in
limit cycle problems.

642
00:45:15 --> 00:45:21
I said yeah,
in particular,

643
00:45:16 --> 00:45:22
I was reading this paper of the
mathematician who found four

644
00:45:20 --> 00:45:26
limit cycles.
And I opened to that system and

645
00:45:23 --> 00:45:29
said the name is,
and I read it out loud.

646
00:45:26 --> 00:45:32
I said do you by any chance
know him?

647
00:45:30 --> 00:45:36
And he smiled and said yes,
very well.

648
00:45:32 --> 00:45:38
That is my mother.
[LAUGHTER]

649
00:45:35 --> 00:45:41

650
00:45:43 --> 00:45:49
Well, bye-bye.