1
00:00:08,000 --> 00:00:14,000
First of all,
the way a nonlinear autonomous
2
00:00:11,000 --> 00:00:17,000
system looks,
you have had some practice with
3
00:00:15,000 --> 00:00:21,000
it by now.
This is nonlinear.
4
00:00:17,000 --> 00:00:23,000
The right-hand side are no
longer simple combinations ax
5
00:00:22,000 --> 00:00:28,000
plus by.
Nonlinear and autonomous,
6
00:00:26,000 --> 00:00:32,000
these are function just of x
and y.
7
00:00:30,000 --> 00:00:36,000
There is no t on the right-hand
side.
8
00:00:33,000 --> 00:00:39,000
Now, most of today will be
geometric.
9
00:00:37,000 --> 00:00:43,000
The way to get a geometric
picture of that is first by
10
00:00:43,000 --> 00:00:49,000
constructing the velocity field
whose components are the
11
00:00:48,000 --> 00:00:54,000
functions f and g.
This is a velocity field that
12
00:00:53,000 --> 00:00:59,000
gives a picture of the system
and has solutions.
13
00:01:00,000 --> 00:01:06,000
The solutions to the system,
from the point of view of
14
00:01:05,000 --> 00:01:11,000
functions, they would look like
pairs of functions,
15
00:01:09,000 --> 00:01:15,000
x of t, y of t.
16
00:01:12,000 --> 00:01:18,000
But, from the point of view of
geometry, when you plot them as
17
00:01:18,000 --> 00:01:24,000
parametric equations,
they are called trajectories of
18
00:01:23,000 --> 00:01:29,000
the field F, which simply means
that they are curves everywhere
19
00:01:29,000 --> 00:01:35,000
having the right velocity.
So a typical curve would look
20
00:01:34,000 --> 00:01:40,000
like --
There is a trajectory.
21
00:01:38,000 --> 00:01:44,000
And we know it is a trajectory
because at each point the vector
22
00:01:42,000 --> 00:01:48,000
on it has, of course,
the right direction,
23
00:01:45,000 --> 00:01:51,000
the tangent direction,
but more than that,
24
00:01:48,000 --> 00:01:54,000
it has the right velocity.
So here, for example,
25
00:01:51,000 --> 00:01:57,000
the point is traveling more
slowly.
26
00:01:53,000 --> 00:01:59,000
Here it is traveling more
rapidly because the velocity
27
00:01:57,000 --> 00:02:03,000
vector is bigger,
longer.
28
00:02:00,000 --> 00:02:06,000
So this is a picture of a
typical trajectory.
29
00:02:03,000 --> 00:02:09,000
The only other things that I
should mention are the critical
30
00:02:09,000 --> 00:02:15,000
points.
If you have worked the problems
31
00:02:12,000 --> 00:02:18,000
for this week,
the first couple of problems,
32
00:02:16,000 --> 00:02:22,000
you have already seen the
significance of the critical
33
00:02:21,000 --> 00:02:27,000
points.
Well, from Monday's lecture you
34
00:02:24,000 --> 00:02:30,000
know from the point of view of
solutions they are constant
35
00:02:29,000 --> 00:02:35,000
solutions.
36
00:02:37,000 --> 00:02:43,000
From the point of view of the
field they are where the field
37
00:02:41,000 --> 00:02:47,000
is zero.
There is no velocity vector,
38
00:02:43,000 --> 00:02:49,000
in other words.
The velocity vector is zero.
39
00:02:46,000 --> 00:02:52,000
And, therefore,
a point being there has no
40
00:02:49,000 --> 00:02:55,000
reason to go anywhere else.
And, spelling it out,
41
00:02:53,000 --> 00:02:59,000
it's where the partial
derivatives, where the values of
42
00:02:57,000 --> 00:03:03,000
the functions on the right-hand
side, which give the two
43
00:03:01,000 --> 00:03:07,000
components, the i and j
components of the field,
44
00:03:04,000 --> 00:03:10,000
where they are zero.
45
00:03:11,000 --> 00:03:17,000
That is all I will need by way
of a recall today.
46
00:03:14,000 --> 00:03:20,000
I don't think I will need
anything else.
47
00:03:17,000 --> 00:03:23,000
The topic for today is another
kind of behavior that you have
48
00:03:22,000 --> 00:03:28,000
not yet observed at the computer
screen, unless you have worked
49
00:03:26,000 --> 00:03:32,000
ahead, and that is there are
trajectories which go along to
50
00:03:31,000 --> 00:03:37,000
infinity or end up at a critical
point.
51
00:03:35,000 --> 00:03:41,000
They are the critical points
that just sit there all the
52
00:03:39,000 --> 00:03:45,000
time.
But there is a third type of
53
00:03:42,000 --> 00:03:48,000
behavior that a trajectory can
have where it neither sits for
54
00:03:47,000 --> 00:03:53,000
all time nor goes off for all
time.
55
00:03:50,000 --> 00:03:56,000
Instead, it repeats itself.
Such a thing is called a closed
56
00:03:55,000 --> 00:04:01,000
trajectory.
What does it look like?
57
00:03:59,000 --> 00:04:05,000
Well, it is a closed curve in
the plane that at every point,
58
00:04:05,000 --> 00:04:11,000
it is a trajectory,
i.e., the arrows at each point,
59
00:04:10,000 --> 00:04:16,000
let's say it is traced in the
clockwise direction.
60
00:04:15,000 --> 00:04:21,000
And so the arrows of the field
will go like this.
61
00:04:20,000 --> 00:04:26,000
Here it is going slowly,
here it is very slow and here
62
00:04:25,000 --> 00:04:31,000
it picks up a little speed again
and so on.
63
00:04:31,000 --> 00:04:37,000
Now, for such a trajectory what
is happening?
64
00:04:35,000 --> 00:04:41,000
Well, it goes around in finite
time and then repeats itself.
65
00:04:42,000 --> 00:04:48,000
It just goes round and round
forever if you land on that
66
00:04:48,000 --> 00:04:54,000
trajectory.
It represents a system that
67
00:04:53,000 --> 00:04:59,000
returns to its original state
periodically.
68
00:04:57,000 --> 00:05:03,000
It represents periodic behavior
of the system.
69
00:05:16,000 --> 00:05:22,000
Now, we have seen one example
of that, a simple example where
70
00:05:23,000 --> 00:05:29,000
this simple system,
x prime equals y,
71
00:05:29,000 --> 00:05:35,000
y prime equals negative x.
72
00:05:40,000 --> 00:05:46,000
We could write down the
solutions to that directly,
73
00:05:43,000 --> 00:05:49,000
but if you want to do
eigenvalues and eigenvectors the
74
00:05:47,000 --> 00:05:53,000
matrix will look like this.
The equation will be lambda
75
00:05:51,000 --> 00:05:57,000
squared plus zero lambda plus
one equals zero,
76
00:05:55,000 --> 00:06:01,000
so the eigenvalues will be plus
77
00:05:58,000 --> 00:06:04,000
or minus i.
In fact, from then on you could
78
00:06:02,000 --> 00:06:08,000
work out in the usual ways the
eigenvectors,
79
00:06:04,000 --> 00:06:10,000
complex eigenvectors and
separate them.
80
00:06:07,000 --> 00:06:13,000
But, look, you can avoid all
that just by writing down the
81
00:06:11,000 --> 00:06:17,000
solution.
The solutions are sines and
82
00:06:13,000 --> 00:06:19,000
cosines.
One basic solution will be x
83
00:06:16,000 --> 00:06:22,000
equals cosine t,
in which case what is y?
84
00:06:19,000 --> 00:06:25,000
Well, y is the derivative of
that.
85
00:06:22,000 --> 00:06:28,000
That will be minus sine t.
86
00:06:24,000 --> 00:06:30,000
Another basic solution,
we will start with x equals
87
00:06:28,000 --> 00:06:34,000
sine t.
In which case y will be cosine
88
00:06:33,000 --> 00:06:39,000
t, its derivative.
89
00:06:36,000 --> 00:06:42,000
Now, if you do that,
what do these things look like?
90
00:06:41,000 --> 00:06:47,000
Well, either of these two basic
solutions looks like a circle,
91
00:06:47,000 --> 00:06:53,000
not traced in the usual way but
in the opposite way.
92
00:06:52,000 --> 00:06:58,000
For example,
when t is equal to zero it
93
00:06:56,000 --> 00:07:02,000
starts at the point one,
zero.
94
00:07:00,000 --> 00:07:06,000
Now, if the minus sign were not
there this would be x equals
95
00:07:04,000 --> 00:07:10,000
cosine t,
y equals sine t,
96
00:07:08,000 --> 00:07:14,000
which is the usual
counterclockwise circle.
97
00:07:11,000 --> 00:07:17,000
But if I change y from sine t
to negative sine t
98
00:07:15,000 --> 00:07:21,000
it is going around the
other way.
99
00:07:18,000 --> 00:07:24,000
So this circle is traced that
way.
100
00:07:20,000 --> 00:07:26,000
And this is a family of
circles, according to the values
101
00:07:24,000 --> 00:07:30,000
of c1 and c2,
concentric, all of which go
102
00:07:27,000 --> 00:07:33,000
around clockwise.
So those are closed
103
00:07:31,000 --> 00:07:37,000
trajectories.
Those are the solutions.
104
00:07:33,000 --> 00:07:39,000
They are trajectories of the
vector field.
105
00:07:36,000 --> 00:07:42,000
They are closed.
They come around and they
106
00:07:39,000 --> 00:07:45,000
repeat in finite time.
Now, these are no good.
107
00:07:42,000 --> 00:07:48,000
These are the kind I am not
interested in.
108
00:07:45,000 --> 00:07:51,000
These are commonplace,
and we are interested in good
109
00:07:49,000 --> 00:07:55,000
stuff today.
And the good stuff we are
110
00:07:51,000 --> 00:07:57,000
interested in is limit cycles.
111
00:08:01,000 --> 00:08:07,000
A limit cycle is a closed
trajectory with a couple of
112
00:08:06,000 --> 00:08:12,000
extra hypotheses.
It is a closed trajectory,
113
00:08:10,000 --> 00:08:16,000
just like those guys,
but it has something they don't
114
00:08:15,000 --> 00:08:21,000
have, namely,
it is king of the roost.
115
00:08:19,000 --> 00:08:25,000
They have to be isolated,
no other guys nearby.
116
00:08:23,000 --> 00:08:29,000
And they also have to be
stable.
117
00:08:27,000 --> 00:08:33,000
See, the problem here is that
none of these stands out from
118
00:08:32,000 --> 00:08:38,000
any of the others.
In other words,
119
00:08:37,000 --> 00:08:43,000
there must be,
isolated means,
120
00:08:40,000 --> 00:08:46,000
no others nearby.
121
00:08:49,000 --> 00:08:55,000
That is just what goes wrong
here.
122
00:08:51,000 --> 00:08:57,000
Arbitrarily close to each of
these circles is yet another
123
00:08:55,000 --> 00:09:01,000
circle doing exactly the same
thing.
124
00:08:58,000 --> 00:09:04,000
That means that there are some
that are only of mild interest.
125
00:09:03,000 --> 00:09:09,000
What is much more interesting
is to find a cycle where there
126
00:09:07,000 --> 00:09:13,000
is nothing nearby.
Something, therefore,
127
00:09:10,000 --> 00:09:16,000
that looks like this.
128
00:09:22,000 --> 00:09:28,000
Here is our pink guy.
Let's make this one go
129
00:09:25,000 --> 00:09:31,000
counterclockwise.
Here is a limit cycle,
130
00:09:28,000 --> 00:09:34,000
it seems to be.
And now what do nearby guys do?
131
00:09:33,000 --> 00:09:39,000
Well, they should approach it.
Somebody here like that does
132
00:09:38,000 --> 00:09:44,000
this, spirals in and gets ever
and every closer to that thing.
133
00:09:44,000 --> 00:09:50,000
Now, it can never join it
because, if it joined it at the
134
00:09:50,000 --> 00:09:56,000
joining point,
I would have two solutions
135
00:09:54,000 --> 00:10:00,000
going through this point.
And that is illegal.
136
00:10:00,000 --> 00:10:06,000
All it can do is get
arbitrarily close.
137
00:10:02,000 --> 00:10:08,000
On the computer screen it will
look as if it joins it but,
138
00:10:06,000 --> 00:10:12,000
of course, it cannot.
It is just the resolution,
139
00:10:10,000 --> 00:10:16,000
the pixels.
Not enough pixels.
140
00:10:12,000 --> 00:10:18,000
The resolution isn't good
enough.
141
00:10:14,000 --> 00:10:20,000
And the ones that start further
away will take longer to find
142
00:10:18,000 --> 00:10:24,000
their way to the limit cycle and
they will always stay outside of
143
00:10:23,000 --> 00:10:29,000
the earlier guys,
but they will get arbitrarily
144
00:10:26,000 --> 00:10:32,000
close, too.
How about inside?
145
00:10:30,000 --> 00:10:36,000
Inside, well,
it starts somewhere and does
146
00:10:33,000 --> 00:10:39,000
the same thing.
It starts here and will try to
147
00:10:37,000 --> 00:10:43,000
join the limit cycle.
That is what I mean by
148
00:10:41,000 --> 00:10:47,000
stability.
Stability means that nearby
149
00:10:44,000 --> 00:10:50,000
guys, the guys that start
somewhere else eventually
150
00:10:48,000 --> 00:10:54,000
approach the limit cycle,
regardless of whether they
151
00:10:52,000 --> 00:10:58,000
start from the outside or start
from the inside.
152
00:10:56,000 --> 00:11:02,000
So that is stable.
An unstable limit cycle --
153
00:11:02,000 --> 00:11:08,000
But I am not calling it a limit
cycle if it is unstable.
154
00:11:06,000 --> 00:11:12,000
I am just calling it a closed
trajectory, but let's draw one
155
00:11:10,000 --> 00:11:16,000
which is unstable.
Here is the way we will look if
156
00:11:14,000 --> 00:11:20,000
it is unstable.
Guys that start nearby will be
157
00:11:17,000 --> 00:11:23,000
repelled, driven somewhere else.
Or, if they start here,
158
00:11:21,000 --> 00:11:27,000
they will go away from the
thing instead of going toward
159
00:11:25,000 --> 00:11:31,000
it.
This is unstable.
160
00:11:27,000 --> 00:11:33,000
And I don't call it a limit
cycle.
161
00:11:29,000 --> 00:11:35,000
It is just a closed trajectory.
162
00:11:38,000 --> 00:11:44,000
Cycle because it cycles round
and round.
163
00:11:40,000 --> 00:11:46,000
Limit because it is the limit
of the nearby curves.
164
00:11:44,000 --> 00:11:50,000
The other case where it is
unstable is not the limit.
165
00:11:48,000 --> 00:11:54,000
Of course, you could have a
case also where the curves
166
00:11:52,000 --> 00:11:58,000
outside spiral in toward it but
the ones inside are repelled and
167
00:11:57,000 --> 00:12:03,000
do this.
That would be called
168
00:11:59,000 --> 00:12:05,000
semi-stable.
And you can make up all sorts
169
00:12:03,000 --> 00:12:09,000
of cases.
And I think I,
170
00:12:05,000 --> 00:12:11,000
at one point,
drew them in the notes,
171
00:12:07,000 --> 00:12:13,000
but I am not going to.
The only interesting one,
172
00:12:11,000 --> 00:12:17,000
of permanent importance that
people study,
173
00:12:14,000 --> 00:12:20,000
are the actual limit cycles.
No, it was the stable closed
174
00:12:19,000 --> 00:12:25,000
trajectories.
Notice, by the way,
175
00:12:21,000 --> 00:12:27,000
a closed trajectory is always a
simple curve.
176
00:12:25,000 --> 00:12:31,000
Remember what that means from
18.02?
177
00:12:29,000 --> 00:12:35,000
Simple means it doesn't cross
itself.
178
00:12:32,000 --> 00:12:38,000
Why doesn't it cross itself?
It cannot cross itself because,
179
00:12:37,000 --> 00:12:43,000
if it tried to,
what is wrong with that point?
180
00:12:41,000 --> 00:12:47,000
At that point which way does
the vector field go,
181
00:12:46,000 --> 00:12:52,000
that way or that way?
Why the interest of limit
182
00:12:50,000 --> 00:12:56,000
cycles?
Well, because there are systems
183
00:12:54,000 --> 00:13:00,000
in nature in which just this
type of behavior,
184
00:12:58,000 --> 00:13:04,000
they have a certain periodic
motion.
185
00:13:03,000 --> 00:13:09,000
And, if you disturb it,
gradually it returns to its
186
00:13:07,000 --> 00:13:13,000
original periodic state.
A simple example is breathing.
187
00:13:11,000 --> 00:13:17,000
Now I have made you all
self-conscious.
188
00:13:14,000 --> 00:13:20,000
All of you are breathing.
If you are here you are
189
00:13:18,000 --> 00:13:24,000
breathing.
At what rate are you breathing?
190
00:13:22,000 --> 00:13:28,000
Well, you are unaware of it,
of course, except now.
191
00:13:26,000 --> 00:13:32,000
If you are sitting here
listening.
192
00:13:30,000 --> 00:13:36,000
There is a certain temperature
and a certain air circulation in
193
00:13:34,000 --> 00:13:40,000
the room.
You are not thinking of
194
00:13:36,000 --> 00:13:42,000
anything, certainly not of the
lecture, and the lecture is not
195
00:13:40,000 --> 00:13:46,000
unduly exciting,
you will breathe at a certain
196
00:13:43,000 --> 00:13:49,000
steady rate which is a little
different for every person but
197
00:13:47,000 --> 00:13:53,000
that is your rate.
Now, you can artificially
198
00:13:50,000 --> 00:13:56,000
change that.
You could say now I am going to
199
00:13:53,000 --> 00:13:59,000
breathe faster.
And indeed you can.
200
00:13:57,000 --> 00:14:03,000
But, as soon as you stop being
aware of what you are doing,
201
00:14:01,000 --> 00:14:07,000
the levels of various hormones
and carbon dioxide in your
202
00:14:06,000 --> 00:14:12,000
bloodstream and so on will
return your breathing to its
203
00:14:11,000 --> 00:14:17,000
natural state.
In other words,
204
00:14:13,000 --> 00:14:19,000
that system of your breathing,
which is controlled by various
205
00:14:18,000 --> 00:14:24,000
chemicals and hormones in the
body, is exhibiting exactly this
206
00:14:23,000 --> 00:14:29,000
type of behavior.
It has a certain regular
207
00:14:27,000 --> 00:14:33,000
periodic motion as a system.
And, if disturbed,
208
00:14:32,000 --> 00:14:38,000
if artificially you set it out
somewhere else,
209
00:14:34,000 --> 00:14:40,000
it will gradually return to its
original state.
210
00:14:37,000 --> 00:14:43,000
Now, of course,
if I am running it will be
211
00:14:40,000 --> 00:14:46,000
different.
Sure.
212
00:14:41,000 --> 00:14:47,000
If you are running you breathe
faster, but that is because the
213
00:14:45,000 --> 00:14:51,000
parameters in the system,
the a's and the b's in the
214
00:14:48,000 --> 00:14:54,000
equation, the f of (x,
y) and g of (x,
215
00:14:50,000 --> 00:14:56,000
y), the parameters in those
216
00:14:53,000 --> 00:14:59,000
functions will be set at
different levels.
217
00:14:56,000 --> 00:15:02,000
You will have different
hormones, a different of carbon
218
00:14:59,000 --> 00:15:05,000
dioxide and so on.
Now, I am not saying that
219
00:15:04,000 --> 00:15:10,000
breathing is modeled by a limit
cycle.
220
00:15:07,000 --> 00:15:13,000
It is the sort of thing which
one might look for a limit
221
00:15:11,000 --> 00:15:17,000
cycle.
That is, of course,
222
00:15:13,000 --> 00:15:19,000
a question for biologists.
And, in general,
223
00:15:17,000 --> 00:15:23,000
any type of periodic behavior
in nature, people try to see if
224
00:15:22,000 --> 00:15:28,000
there is some system of
differential equations which
225
00:15:26,000 --> 00:15:32,000
governs it in which perhaps
there is a limit cycle,
226
00:15:30,000 --> 00:15:36,000
which contains a limit cycle.
Well, what are the problems?
227
00:15:36,000 --> 00:15:42,000
In a sense, limit cycles are
easy to lecture about because so
228
00:15:41,000 --> 00:15:47,000
little is known about them.
At the end of the period,
229
00:15:46,000 --> 00:15:52,000
if I have time,
I will show you that the
230
00:15:49,000 --> 00:15:55,000
simplest possible question you
could ask, the answer to it is
231
00:15:55,000 --> 00:16:01,000
totally known after 120 years of
steady trying.
232
00:16:00,000 --> 00:16:06,000
But let's first talk about what
sorts of problems people address
233
00:16:05,000 --> 00:16:11,000
with limit cycles.
First of all is the existence
234
00:16:09,000 --> 00:16:15,000
problem.
235
00:16:16,000 --> 00:16:22,000
If I give you a system,
you know, the right-hand side
236
00:16:19,000 --> 00:16:25,000
is x squared plus 2y cubed minus
3xy,
237
00:16:23,000 --> 00:16:29,000
and the g is something similar.
I say does this have limit
238
00:16:26,000 --> 00:16:32,000
cycles?
Well, you know how to find its
239
00:16:29,000 --> 00:16:35,000
critical points.
But how do you find out if it
240
00:16:34,000 --> 00:16:40,000
has limit cycles?
The answer to that is nobody
241
00:16:40,000 --> 00:16:46,000
has any idea.
This problem,
242
00:16:44,000 --> 00:16:50,000
in general, there are not much
in the way of methods.
243
00:16:50,000 --> 00:16:56,000
Not much.
244
00:17:00,000 --> 00:17:06,000
Not much is known.
There is one theorem that you
245
00:17:03,000 --> 00:17:09,000
will find in the notes,
a simple theorem called the
246
00:17:07,000 --> 00:17:13,000
Poincare-Bendixson theorem
which, for about 60 or 70 years
247
00:17:11,000 --> 00:17:17,000
was about the only result known
which enabled people to find
248
00:17:16,000 --> 00:17:22,000
limit cycles.
Nowadays the theorem is used
249
00:17:19,000 --> 00:17:25,000
relatively little because people
try to find limit cycles by
250
00:17:24,000 --> 00:17:30,000
computer.
Now, the difficulty is you have
251
00:17:27,000 --> 00:17:33,000
to know where to look for them.
In other words,
252
00:17:32,000 --> 00:17:38,000
the computer screen shows that
much and you set the axes and it
253
00:17:36,000 --> 00:17:42,000
doesn't show any limit cycles.
That doesn't mean there are not
254
00:17:40,000 --> 00:17:46,000
any.
That means they are over there,
255
00:17:43,000 --> 00:17:49,000
or it means there is a big one
like there.
256
00:17:46,000 --> 00:17:52,000
And you are looking in the
middle of it and don't see it.
257
00:17:50,000 --> 00:17:56,000
So, in general,
people don't look for limit
258
00:17:53,000 --> 00:17:59,000
cycles unless the physical
system that gave rise to the
259
00:17:56,000 --> 00:18:02,000
pair of differential equations
suggests that there is something
260
00:18:01,000 --> 00:18:07,000
repetitive going on like
breathing.
261
00:18:05,000 --> 00:18:11,000
And, if it tells you that,
then it often gives you
262
00:18:09,000 --> 00:18:15,000
approximate values of the
parameters and the variables so
263
00:18:14,000 --> 00:18:20,000
you know where to look.
Basically this is done by
264
00:18:18,000 --> 00:18:24,000
computer search guided by the
physical problem.
265
00:18:31,000 --> 00:18:37,000
Therefore, I cannot say much
more about it today.
266
00:18:35,000 --> 00:18:41,000
Instead I am going to focus my
attention on nonexistence.
267
00:18:40,000 --> 00:18:46,000
When can you be sure that a
system will not have any limit
268
00:18:46,000 --> 00:18:52,000
cycles?
And there are two theorems.
269
00:18:49,000 --> 00:18:55,000
One, again, due to Bendixson
who was a Swedish mathematician
270
00:18:54,000 --> 00:19:00,000
who lived around 1900 or so.
There is a criterion due to
271
00:18:59,000 --> 00:19:05,000
Bendixson.
And there is one involving
272
00:19:04,000 --> 00:19:10,000
critical points.
And I would like to describe
273
00:19:08,000 --> 00:19:14,000
both of them for you today.
First of all,
274
00:19:11,000 --> 00:19:17,000
Bendixson's criterion.
275
00:19:22,000 --> 00:19:28,000
It is very simply stated and
has a marvelous proof,
276
00:19:25,000 --> 00:19:31,000
which I am going to give you.
We have D as a region of the
277
00:19:29,000 --> 00:19:35,000
plane.
278
00:19:35,000 --> 00:19:41,000
And what Bendixson's criterion
tells you to do is take your
279
00:19:40,000 --> 00:19:46,000
vector field and calculate its
divergence.
280
00:19:43,000 --> 00:19:49,000
We are set back in 1802,
and this proof is going to be
281
00:19:48,000 --> 00:19:54,000
straight 18.02.
You will enjoy it.
282
00:19:51,000 --> 00:19:57,000
Calculate the divergence.
Now, I am talking about the
283
00:19:55,000 --> 00:20:01,000
two-dimensional divergence.
Remember that is fx,
284
00:20:00,000 --> 00:20:06,000
the partial of f with respect
to x, plus the partial of the g,
285
00:20:05,000 --> 00:20:11,000
the j component with respect to
y.
286
00:20:08,000 --> 00:20:14,000
And assume that that is a
continuous function.
287
00:20:12,000 --> 00:20:18,000
It always will be with us.
Practically all the examples I
288
00:20:16,000 --> 00:20:22,000
will give you f and g will be
simple polynomials.
289
00:20:20,000 --> 00:20:26,000
They are smooth,
continuous and nice and behave
290
00:20:24,000 --> 00:20:30,000
as you want.
And you calculate that and
291
00:20:27,000 --> 00:20:33,000
assume --
Suppose, in other words,
292
00:20:32,000 --> 00:20:38,000
that the divergence of f,
I need more room.
293
00:20:36,000 --> 00:20:42,000
The hypothesis is that the
divergence of f is not zero in
294
00:20:42,000 --> 00:20:48,000
that region D.
It is never zero.
295
00:20:45,000 --> 00:20:51,000
It is not zero at any point in
that region.
296
00:20:49,000 --> 00:20:55,000
The conclusion is that there
are no limit cycles in the
297
00:20:55,000 --> 00:21:01,000
region.
If it is not zero in D,
298
00:20:58,000 --> 00:21:04,000
there are no limit cycles.
In fact, there are not even any
299
00:21:04,000 --> 00:21:10,000
closed trajectories.
You couldn't even have those
300
00:21:09,000 --> 00:21:15,000
bunch of concentric circles,
so there are no closed
301
00:21:13,000 --> 00:21:19,000
trajectories of the original
system whose divergence you
302
00:21:18,000 --> 00:21:24,000
calculated.
There are no closed
303
00:21:21,000 --> 00:21:27,000
trajectories in D.
For example,
304
00:21:24,000 --> 00:21:30,000
let me give you a simple
example to put a little flesh on
305
00:21:29,000 --> 00:21:35,000
it.
Let's see.
306
00:21:32,000 --> 00:21:38,000
What do I have?
I prepared an example.
307
00:21:35,000 --> 00:21:41,000
x prime equals,
here is a simple nonlinear
308
00:21:40,000 --> 00:21:46,000
system, x cubed plus y cubed.
309
00:21:45,000 --> 00:21:51,000
And y prime equals 3x plus y
cubed plus 2y.
310
00:21:59,000 --> 00:22:05,000
Does this system have limit
cycles?
311
00:22:01,000 --> 00:22:07,000
Well, even to calculate its
critical points would be a
312
00:22:05,000 --> 00:22:11,000
little task, but we can easily
answer the question as to
313
00:22:09,000 --> 00:22:15,000
whether it has limit cycles or
not by Bendixson's criterion.
314
00:22:14,000 --> 00:22:20,000
Let's calculate the divergence.
The divergence of the vector
315
00:22:18,000 --> 00:22:24,000
field whose components are these
two functions is,
316
00:22:22,000 --> 00:22:28,000
well, 3x squared,
it's the partial of the first
317
00:22:26,000 --> 00:22:32,000
guy with respect to x plus the
partial of the second guy with
318
00:22:31,000 --> 00:22:37,000
respect to y,
which is 3y squared plus two.
319
00:22:34,000 --> 00:22:40,000
Now, can that be zero anywhere
320
00:22:39,000 --> 00:22:45,000
in the x,y-plane?
No, because it is the sum of
321
00:22:43,000 --> 00:22:49,000
these two squares.
This much of it could be zero
322
00:22:47,000 --> 00:22:53,000
only at the origin,
but that plus two eliminates
323
00:22:51,000 --> 00:22:57,000
even that.
This is always positive in the
324
00:22:55,000 --> 00:23:01,000
entire x,y-plane.
Here my domain is the whole
325
00:22:59,000 --> 00:23:05,000
x,y-plane and,
therefore, the conclusion is
326
00:23:03,000 --> 00:23:09,000
that there are no closed
trajectories in the x,y-plane,
327
00:23:07,000 --> 00:23:13,000
anywhere.
328
00:23:15,000 --> 00:23:21,000
And we have done that with just
a couple of lines of calculation
329
00:23:18,000 --> 00:23:24,000
and nothing further required.
No computer search.
330
00:23:21,000 --> 00:23:27,000
In fact, no computer search
could ever proof this.
331
00:23:24,000 --> 00:23:30,000
It would be impossible because,
no matter where you look,
332
00:23:27,000 --> 00:23:33,000
there is always some other
place to look.
333
00:23:30,000 --> 00:23:36,000
This is an example where a
couple lines of mathematics
334
00:23:35,000 --> 00:23:41,000
dispose of the matter far more
effectively than a million
335
00:23:41,000 --> 00:23:47,000
dollars worth of calculation.
Well, where does Bendixson's
336
00:23:46,000 --> 00:23:52,000
theorem come from?
Yes, Bendixson's theorem comes
337
00:23:51,000 --> 00:23:57,000
from 18.02.
And I am giving it to you both
338
00:23:56,000 --> 00:24:02,000
to recall a little bit of 18.02
to you.
339
00:24:01,000 --> 00:24:07,000
Because it is about the first
example in the course that we
340
00:24:06,000 --> 00:24:12,000
have had of an indirect
argument.
341
00:24:09,000 --> 00:24:15,000
And indirect arguments are
something you have to slowly get
342
00:24:14,000 --> 00:24:20,000
used to.
I am going to give you an
343
00:24:17,000 --> 00:24:23,000
indirect proof.
Remember what that is?
344
00:24:20,000 --> 00:24:26,000
You assume the contrary and you
show it leads to a
345
00:24:25,000 --> 00:24:31,000
contradiction.
What would assuming the
346
00:24:28,000 --> 00:24:34,000
contrary be?
Contrary would be I will assume
347
00:24:34,000 --> 00:24:40,000
the divergence is not zero,
but I will suppose there is a
348
00:24:40,000 --> 00:24:46,000
closed trajectory.
Suppose there is a closed
349
00:24:44,000 --> 00:24:50,000
trajectory that exists.
350
00:24:56,000 --> 00:25:02,000
Let's draw a picture of it.
351
00:25:08,000 --> 00:25:14,000
And let's say it goes around
this way.
352
00:25:11,000 --> 00:25:17,000
There is a closed trajectory
for our system.
353
00:25:15,000 --> 00:25:21,000
Let's call the curve C.
And I am going to call the
354
00:25:19,000 --> 00:25:25,000
inside of it R,
the way one often does in
355
00:25:22,000 --> 00:25:28,000
18.02.
D is all this region out here,
356
00:25:25,000 --> 00:25:31,000
in which everything is taking
place.
357
00:25:30,000 --> 00:25:36,000
This is to exist in D.
Now, what I am going to do is
358
00:25:35,000 --> 00:25:41,000
calculate a line integral around
that curve.
359
00:25:45,000 --> 00:25:51,000
A line integral of this vector
field.
360
00:25:47,000 --> 00:25:53,000
Now, there are two things you
can calculate.
361
00:25:51,000 --> 00:25:57,000
One of the line integrals,
I will put in a few of the
362
00:25:55,000 --> 00:26:01,000
vectors here.
The vectors I know are pointing
363
00:25:58,000 --> 00:26:04,000
this way because that is the
direction in which the curve is
364
00:26:03,000 --> 00:26:09,000
being traversed in order to make
it a trajectory.
365
00:26:08,000 --> 00:26:14,000
Those are a few of the typical
vectors in the field.
366
00:26:12,000 --> 00:26:18,000
I am going to calculate the
line integral around that curve
367
00:26:16,000 --> 00:26:22,000
in the positive sense.
In other words,
368
00:26:19,000 --> 00:26:25,000
not in the direction of the
salmon-colored arrow,
369
00:26:23,000 --> 00:26:29,000
but in the normal sense in
which you calculate it using
370
00:26:27,000 --> 00:26:33,000
Green's theorem,
for example.
371
00:26:31,000 --> 00:26:37,000
The positive sense means the
one which keeps the region,
372
00:26:34,000 --> 00:26:40,000
the inside on your left,
as you walk around like that
373
00:26:38,000 --> 00:26:44,000
the region stays on your left.
That is the positive sense.
374
00:26:42,000 --> 00:26:48,000
That is the sense in which I am
integrating.
375
00:26:45,000 --> 00:26:51,000
I am going to use Green's
theorem, but the integral that I
376
00:26:49,000 --> 00:26:55,000
am going to calculate is not the
work integral.
377
00:26:52,000 --> 00:26:58,000
I am going to calculate instead
the flux integral,
378
00:26:55,000 --> 00:27:01,000
the integral that represents
the flux of F across C.
379
00:27:00,000 --> 00:27:06,000
Now, what is that integral?
Well, at each point,
380
00:27:04,000 --> 00:27:10,000
you station a little ant and
the ant reports the outward flow
381
00:27:09,000 --> 00:27:15,000
rate across that point which is
F dotted with the normal vector.
382
00:27:15,000 --> 00:27:21,000
I will put in a few normal
vectors just to remind you.
383
00:27:20,000 --> 00:27:26,000
The normal vectors look like
little unit vectors pointing
384
00:27:25,000 --> 00:27:31,000
perpendicularly outwards
everywhere.
385
00:27:29,000 --> 00:27:35,000
These are the n's.
F dotted with the unit normal
386
00:27:34,000 --> 00:27:40,000
vector, and that is added up
around the curve.
387
00:27:38,000 --> 00:27:44,000
This quantity gives me the flux
of the field across C.
388
00:27:43,000 --> 00:27:49,000
Now, we are going to calculate
that by Green's theorem.
389
00:27:48,000 --> 00:27:54,000
But, before we calculate it by
Green's theorem,
390
00:27:52,000 --> 00:27:58,000
we are going to psych it out.
What is it?
391
00:27:55,000 --> 00:28:01,000
What is the value of that
integral?
392
00:28:00,000 --> 00:28:06,000
Well, since I am asking you to
do it in your head there can
393
00:28:04,000 --> 00:28:10,000
only be one possible answer.
It is zero.
394
00:28:06,000 --> 00:28:12,000
Why is that integral zero?
Well, because at each point the
395
00:28:10,000 --> 00:28:16,000
field vector,
the velocity vector is
396
00:28:13,000 --> 00:28:19,000
perpendicular to the normal
vector.
397
00:28:15,000 --> 00:28:21,000
Why?
The normal vector points
398
00:28:17,000 --> 00:28:23,000
perpendicularly to the curve but
the field vector always is
399
00:28:22,000 --> 00:28:28,000
tangent to the curve because
this curve is a trajectory.
400
00:28:27,000 --> 00:28:33,000
It is always supposed to be
going in the direction given by
401
00:28:34,000 --> 00:28:40,000
that white field vector.
Do you follow?
402
00:28:39,000 --> 00:28:45,000
A trajectory means that it is
always tangent to the field
403
00:28:47,000 --> 00:28:53,000
vector and, therefore,
always perpendicular to the
404
00:28:53,000 --> 00:28:59,000
normal vector.
This is zero since F dot n is
405
00:28:59,000 --> 00:29:05,000
always zero.
Everywhere on the curve,
406
00:29:03,000 --> 00:29:09,000
F dot n has to be zero.
There is no flux of this field
407
00:29:07,000 --> 00:29:13,000
across the curve because the
field is always in the same
408
00:29:12,000 --> 00:29:18,000
direction as the curve,
never perpendicular to it.
409
00:29:15,000 --> 00:29:21,000
It has no components
perpendicular to it.
410
00:29:19,000 --> 00:29:25,000
Good.
Now let's do it the hard way.
411
00:29:21,000 --> 00:29:27,000
Let's use Green's theorem.
Green's theorem says that the
412
00:29:25,000 --> 00:29:31,000
flux across C should be equal to
the double integral over that
413
00:29:30,000 --> 00:29:36,000
region of the divergence of F.
It's like Gauss theorem in two
414
00:29:35,000 --> 00:29:41,000
dimensions, this version of it.
Divergence of F,
415
00:29:39,000 --> 00:29:45,000
that is a function,
I double integrate it over the
416
00:29:42,000 --> 00:29:48,000
region, and then that is dx /
dy, or let's say da because you
417
00:29:46,000 --> 00:29:52,000
might want to do it in polar
coordinates.
418
00:29:48,000 --> 00:29:54,000
And, on the problem set,
you certainly will want to do
419
00:29:52,000 --> 00:29:58,000
it in polar coordinates,
I think.
420
00:30:00,000 --> 00:30:06,000
All right.
How much is that?
421
00:30:02,000 --> 00:30:08,000
Well, we haven't yet used the
hypothesis.
422
00:30:06,000 --> 00:30:12,000
All we have done is set up the
problem.
423
00:30:10,000 --> 00:30:16,000
Now, the hypothesis was that
the divergence is never zero
424
00:30:16,000 --> 00:30:22,000
anywhere in D.
Therefore, the divergence is
425
00:30:20,000 --> 00:30:26,000
never zero anywhere in R.
What I say is the divergence is
426
00:30:26,000 --> 00:30:32,000
either greater than zero
everywhere in R.
427
00:30:32,000 --> 00:30:38,000
Or less than zero everywhere in
R.
428
00:30:35,000 --> 00:30:41,000
But it cannot be sometimes
positive and sometimes negative.
429
00:30:40,000 --> 00:30:46,000
Why not?
In other words,
430
00:30:42,000 --> 00:30:48,000
I say it is not possible the
divergence here is one and here
431
00:30:47,000 --> 00:30:53,000
is minus two.
That is not possible because,
432
00:30:51,000 --> 00:30:57,000
if I drew a line from this
point to that,
433
00:30:55,000 --> 00:31:01,000
along that line the divergence
would start positive and end up
434
00:31:00,000 --> 00:31:06,000
negative.
And, therefore,
435
00:31:04,000 --> 00:31:10,000
have to be zero some time in
between.
436
00:31:06,000 --> 00:31:12,000
It's because it is a continuous
function.
437
00:31:09,000 --> 00:31:15,000
It is a continuous function.
I am assuming that.
438
00:31:12,000 --> 00:31:18,000
And, therefore,
if it sometimes positive and
439
00:31:15,000 --> 00:31:21,000
sometimes negative it has to be
zero in between.
440
00:31:18,000 --> 00:31:24,000
You cannot get continuously
from plus one to minus two
441
00:31:22,000 --> 00:31:28,000
without passing through zero.
The reason for this is,
442
00:31:27,000 --> 00:31:33,000
since the divergence is never
zero in R it therefore must
443
00:31:35,000 --> 00:31:41,000
always stay positive or always
stay negative.
444
00:31:40,000 --> 00:31:46,000
Now, if it always stays
positive, the conclusion is then
445
00:31:47,000 --> 00:31:53,000
this double integral must be
positive.
446
00:31:52,000 --> 00:31:58,000
Therefore, this double integral
is either greater than zero.
447
00:32:01,000 --> 00:32:07,000
That is if the divergence is
always positive.
448
00:32:04,000 --> 00:32:10,000
Or, it is less than zero if the
divergence is always negative.
449
00:32:10,000 --> 00:32:16,000
But the one thing it cannot be
is not zero.
450
00:32:14,000 --> 00:32:20,000
Well, the left-hand side,
Green's theorem is supposed to
451
00:32:18,000 --> 00:32:24,000
be true.
Green's theorem is our bedrock.
452
00:32:22,000 --> 00:32:28,000
18.02 would crumble without
that so it must be true.
453
00:32:26,000 --> 00:32:32,000
One way of calculating the
left-hand side gives us zero.
454
00:32:33,000 --> 00:32:39,000
If we calculate the right-hand
side it is not zero.
455
00:32:36,000 --> 00:32:42,000
That is called the
contradiction.
456
00:32:38,000 --> 00:32:44,000
Where did the contradiction
arise from?
457
00:32:41,000 --> 00:32:47,000
It arose from the fact that I
supposed that there was a closed
458
00:32:45,000 --> 00:32:51,000
trajectory in that region.
The conclusion is there cannot
459
00:32:48,000 --> 00:32:54,000
be any closed trajectory of that
region because it leads to a
460
00:32:52,000 --> 00:32:58,000
contradiction via Green's
theorem.
461
00:33:02,000 --> 00:33:08,000
Let me see if I can give you
some of the argument for the
462
00:33:06,000 --> 00:33:12,000
other, well, let's at least
state the other criterion I
463
00:33:10,000 --> 00:33:16,000
wanted to give you.
464
00:33:21,000 --> 00:33:27,000
Suppose, for example,
we use this system,
465
00:33:25,000 --> 00:33:31,000
x prime equals --
466
00:33:50,000 --> 00:33:56,000
Does this have limit cycles?
467
00:33:59,000 --> 00:34:05,000
Does that have limit cycles?
468
00:34:08,000 --> 00:34:14,000
Let's Bendixson it.
We will calculate the
469
00:34:12,000 --> 00:34:18,000
divergence of a vector field.
It is 2x from the top function.
470
00:34:18,000 --> 00:34:24,000
The partial with respect to x
is 2x.
471
00:34:22,000 --> 00:34:28,000
The second function with
respect to y is negative 2y.
472
00:34:29,000 --> 00:34:35,000
That certainly could be zero.
In fact, this is zero along the
473
00:34:34,000 --> 00:34:40,000
entire line x equals y.
Its divergence is zero here
474
00:34:39,000 --> 00:34:45,000
along that whole line.
The best I could conclude was,
475
00:34:44,000 --> 00:34:50,000
I could conclude that there is
no limit cycle like this and
476
00:34:50,000 --> 00:34:56,000
there is no limit cycle like
this, but there is nothing so
477
00:34:55,000 --> 00:35:01,000
far that says a limit cycle
could not cross that because
478
00:35:00,000 --> 00:35:06,000
that would not violate
Bendixson's theorem.
479
00:35:06,000 --> 00:35:12,000
In other words,
any domain that contained part
480
00:35:09,000 --> 00:35:15,000
of this line,
the divergence would be zero
481
00:35:13,000 --> 00:35:19,000
along that line.
And, therefore,
482
00:35:15,000 --> 00:35:21,000
I could conclude nothing.
I could have limit cycles that
483
00:35:20,000 --> 00:35:26,000
cross that line,
as long as they included a
484
00:35:23,000 --> 00:35:29,000
piece of that line in them.
The answer is I cannot make a
485
00:35:28,000 --> 00:35:34,000
conclusion.
Well, that is because I am
486
00:35:32,000 --> 00:35:38,000
using the wrong criterion.
Let's instead use the critical
487
00:35:36,000 --> 00:35:42,000
point criterion.
488
00:35:55,000 --> 00:36:01,000
Now, I am going to say that it
makes a nice positive statement
489
00:35:58,000 --> 00:36:04,000
but nobody ever uses it this
way.
490
00:36:01,000 --> 00:36:07,000
Nonetheless,
let's first state it
491
00:36:03,000 --> 00:36:09,000
positively, even though that is
not the way to use it.
492
00:36:07,000 --> 00:36:13,000
The positive statement will be,
once again, we have our region
493
00:36:13,000 --> 00:36:19,000
D and we have a region of the xy
plane and we have our C,
494
00:36:20,000 --> 00:36:26,000
a closed trajectory in it.
A closed trajectory of what?
495
00:36:26,000 --> 00:36:32,000
Of our system.
And that is supposed to be in
496
00:36:30,000 --> 00:36:36,000
D.
The critical point criterion
497
00:36:35,000 --> 00:36:41,000
says something very simple.
If you have that situation it
498
00:36:42,000 --> 00:36:48,000
says that inside that closed
trajectory there must be a
499
00:36:48,000 --> 00:36:54,000
critical point somewhere.
500
00:37:00,000 --> 00:37:06,000
It says that inside C is a
critical point.
501
00:37:15,000 --> 00:37:21,000
Now, this won't help us with
the existence problem.
502
00:37:18,000 --> 00:37:24,000
This won't help us find a
closed trajectory.
503
00:37:21,000 --> 00:37:27,000
We will take our system and say
it has a critical point here and
504
00:37:26,000 --> 00:37:32,000
a critical point there.
Does it have a closed
505
00:37:29,000 --> 00:37:35,000
trajectory?
Well, all I know is the closed
506
00:37:33,000 --> 00:37:39,000
trajectory, if it exists,
will have to go around one or
507
00:37:36,000 --> 00:37:42,000
more of those critical points.
But I don't know where.
508
00:37:40,000 --> 00:37:46,000
It is not going to go around it
like this.
509
00:37:43,000 --> 00:37:49,000
It might go around it like
this.
510
00:37:45,000 --> 00:37:51,000
And my computer search won't
find it because it is looking at
511
00:37:49,000 --> 00:37:55,000
too small a part of the screen.
It doesn't work that way.
512
00:37:53,000 --> 00:37:59,000
It works negatively by
contraposition.
513
00:37:56,000 --> 00:38:02,000
Do you know what the
contrapositive is?
514
00:38:00,000 --> 00:38:06,000
You will at least learn that.
A implies B says the same thing
515
00:38:07,000 --> 00:38:13,000
as not B implies not A.
516
00:38:20,000 --> 00:38:26,000
They are different statements
but they are equivalent to each
517
00:38:24,000 --> 00:38:30,000
other.
If you prove one you prove the
518
00:38:27,000 --> 00:38:33,000
other.
What would be the
519
00:38:29,000 --> 00:38:35,000
contrapositive here?
If you have a closed trajectory
520
00:38:35,000 --> 00:38:41,000
inside is a critical point.
The theorem is used this way.
521
00:38:44,000 --> 00:38:50,000
If D has no critical points,
it has no closed trajectories
522
00:38:53,000 --> 00:38:59,000
and therefore has no limit
cycle.
523
00:39:00,000 --> 00:39:06,000
Because, if it did have a
closed trajectory,
524
00:39:03,000 --> 00:39:09,000
inside it would be a critical
point.
525
00:39:07,000 --> 00:39:13,000
But I said B had no critical
point.
526
00:39:10,000 --> 00:39:16,000
That enables us to dispose of
this system that Bendixson could
527
00:39:15,000 --> 00:39:21,000
not handle at all.
We can dispose of this system
528
00:39:20,000 --> 00:39:26,000
immediately.
Namely, what is it?
529
00:39:22,000 --> 00:39:28,000
Where are its critical points?
Well, where is that zero?
530
00:39:29,000 --> 00:39:35,000
x squared plus y
squared is one,
531
00:39:32,000 --> 00:39:38,000
plus one is never zero.
This is positive.
532
00:39:35,000 --> 00:39:41,000
Or, worse, zero.
And then I add the one to it
533
00:39:38,000 --> 00:39:44,000
and it is not zero anymore.
This has no zeros and,
534
00:39:42,000 --> 00:39:48,000
therefore, it does not matter
that this one has a lot of
535
00:39:46,000 --> 00:39:52,000
zeros.
It makes no difference.
536
00:39:48,000 --> 00:39:54,000
It has no critical points.
It has none,
537
00:39:51,000 --> 00:39:57,000
therefore, no limit cycles.
538
00:40:01,000 --> 00:40:07,000
Now, I desperately wanted to
give you the proof of this.
539
00:40:04,000 --> 00:40:10,000
It is clearly impossible in the
time remaining.
540
00:40:07,000 --> 00:40:13,000
The proof requires a little
time.
541
00:40:09,000 --> 00:40:15,000
I haven't decided what to do
about that.
542
00:40:12,000 --> 00:40:18,000
It might leak over until
Friday's lecture.
543
00:40:15,000 --> 00:40:21,000
Instead, I will finish by
telling you a story.
544
00:40:18,000 --> 00:40:24,000
How is that?
545
00:40:37,000 --> 00:40:43,000
And along side of it was little
y prime.
546
00:40:39,000 --> 00:40:45,000
I am not going to continue on
with the letters of the
547
00:40:43,000 --> 00:40:49,000
alphabet.
I will prime the earlier one.
548
00:40:46,000 --> 00:40:52,000
This has a total of 12
parameters in it.
549
00:40:49,000 --> 00:40:55,000
But, in fact,
if you change variables you can
550
00:40:52,000 --> 00:40:58,000
get rid of all the linear terms.
The important part of it is
551
00:40:57,000 --> 00:41:03,000
only the quadratic terms in the
beginning.
552
00:41:01,000 --> 00:41:07,000
This sort of thing is called a
quadratic system.
553
00:41:05,000 --> 00:41:11,000
After you have departed from
linear systems,
554
00:41:09,000 --> 00:41:15,000
it is the simplest kind there
is.
555
00:41:12,000 --> 00:41:18,000
And the predictor-prey,
the robin-earthworm example I
556
00:41:16,000 --> 00:41:22,000
gave you is of a typical
quadratic system and exhibits
557
00:41:21,000 --> 00:41:27,000
typical nonlinear quadratic
system behavior.
558
00:41:25,000 --> 00:41:31,000
Now, the problem is the
following.
559
00:41:30,000 --> 00:41:36,000
A, b, c, d, e,
f and so on,
560
00:41:32,000 --> 00:41:38,000
those are just real numbers,
parameters, so I am allowed to
561
00:41:37,000 --> 00:41:43,000
give them any values I want.
And the problem that has
562
00:41:42,000 --> 00:41:48,000
bothered people since 1880 when
it was first proposed is how
563
00:41:47,000 --> 00:41:53,000
many limit cycles can a
quadratic system have?
564
00:42:03,000 --> 00:42:09,000
After 120 years this problem is
totally unsolved,
565
00:42:07,000 --> 00:42:13,000
and the mathematicians of the
world who are interested in it
566
00:42:12,000 --> 00:42:18,000
cannot even agree with each
other on what the right
567
00:42:16,000 --> 00:42:22,000
conjecture is.
But let me tell you a little
568
00:42:20,000 --> 00:42:26,000
bit of its history.
There were attempts to solve it
569
00:42:24,000 --> 00:42:30,000
in the 20 or 30 years after it
was first proposed,
570
00:42:28,000 --> 00:42:34,000
through the 1920 and `30s which
all seemed to have gaps in them.
571
00:42:35,000 --> 00:42:41,000
Until finally around 1950 two
Russians mathematicians,
572
00:42:39,000 --> 00:42:45,000
one of whom is extremely
well-known, Petrovski,
573
00:42:44,000 --> 00:42:50,000
a specialist in systems of
ordinary differential equations
574
00:42:49,000 --> 00:42:55,000
published a long and difficult,
complicated 100 page paper in
575
00:42:54,000 --> 00:43:00,000
which they proved that the
maximum number is three.
576
00:43:00,000 --> 00:43:06,000
I won't put down their names.
Petrovski-Landis.
577
00:43:03,000 --> 00:43:09,000
The maximum number was three.
And then not many people were
578
00:43:08,000 --> 00:43:14,000
able to read the paper,
and those who did there seemed
579
00:43:12,000 --> 00:43:18,000
to be gaps in the reasoning in
various places until finally
580
00:43:16,000 --> 00:43:22,000
Arnold who was the greatest
Russian, in my opinion,
581
00:43:20,000 --> 00:43:26,000
one of the greatest Russian
mathematician,
582
00:43:23,000 --> 00:43:29,000
certainly in this field of
analysis and differential
583
00:43:27,000 --> 00:43:33,000
equations, but in other fields,
too, he still is great,
584
00:43:31,000 --> 00:43:37,000
although he is somewhat older
now, criticized it.
585
00:43:37,000 --> 00:43:43,000
He said look,
there is a really big gap in
586
00:43:41,000 --> 00:43:47,000
this argument and it really
cannot be considered to be
587
00:43:46,000 --> 00:43:52,000
proven.
People tried working very hard
588
00:43:50,000 --> 00:43:56,000
to patch it up and without
success.
589
00:43:53,000 --> 00:43:59,000
Then about 1972 or so,
'75 maybe, a Chinese
590
00:43:58,000 --> 00:44:04,000
mathematician found a system
with four.
591
00:44:03,000 --> 00:44:09,000
Wrote down the numbers,
the number a is so much,
592
00:44:07,000 --> 00:44:13,000
b is so much,
and they were absurd numbers
593
00:44:11,000 --> 00:44:17,000
like 10^-6 and 40 billion and so
on, nothing you could plot on a
594
00:44:17,000 --> 00:44:23,000
computer screen,
but found a system with four.
595
00:44:22,000 --> 00:44:28,000
Nobody after this tried to fill
in the gap in the
596
00:44:26,000 --> 00:44:32,000
Petrovski-Landis paper.
I was then chairman of the math
597
00:44:32,000 --> 00:44:38,000
department, and one of my tasks
was protocol and so on.
598
00:44:36,000 --> 00:44:42,000
Anyway, we were trying very
hard to attract a Chinese
599
00:44:40,000 --> 00:44:46,000
mathematician to our department
to become a full professor.
600
00:44:44,000 --> 00:44:50,000
He was a really outstanding
analyst and specialist in
601
00:44:48,000 --> 00:44:54,000
various fields.
Anyway, he came in for a
602
00:44:50,000 --> 00:44:56,000
courtesy interview and we
chatted.
603
00:44:53,000 --> 00:44:59,000
At the time,
I was very much interested in
604
00:44:56,000 --> 00:45:02,000
limit cycles.
And I had on my desk the Math
605
00:44:59,000 --> 00:45:05,000
Society's translation of the
Chinese book on limit cycles.
606
00:45:05,000 --> 00:45:11,000
A collection of papers by
Chinese mathematicians all on
607
00:45:08,000 --> 00:45:14,000
limit cycles.
After a certain point he said,
608
00:45:11,000 --> 00:45:17,000
oh, I see you're interested in
limit cycle problems.
609
00:45:15,000 --> 00:45:21,000
I said yeah,
in particular,
610
00:45:16,000 --> 00:45:22,000
I was reading this paper of the
mathematician who found four
611
00:45:20,000 --> 00:45:26,000
limit cycles.
And I opened to that system and
612
00:45:23,000 --> 00:45:29,000
said the name is,
and I read it out loud.
613
00:45:26,000 --> 00:45:32,000
I said do you by any chance
know him?
614
00:45:30,000 --> 00:45:36,000
And he smiled and said yes,
very well.
615
00:45:32,000 --> 00:45:38,000
That is my mother.
[LAUGHTER]
616
00:45:43,000 --> 00:45:49,000
Well, bye-bye.