1
00:00:01,009 --> 00:00:01,800
GILBERT STRANG: OK.

2
00:00:01,800 --> 00:00:06,540
I'm concentrating now on the
key question of stability.

3
00:00:06,540 --> 00:00:14,320
Do the solutions approach 0 in
the case of linear equations?

4
00:00:14,320 --> 00:00:19,570
Do they approach some constant,
some steady state in the case

5
00:00:19,570 --> 00:00:22,120
of non-linear equations?

6
00:00:22,120 --> 00:00:25,550
So today is the
beginning of non-linear.

7
00:00:25,550 --> 00:00:28,360
I'll start with one equation.

8
00:00:28,360 --> 00:00:34,810
dy dt is some function of y,
not a linear function probably.

9
00:00:34,810 --> 00:00:41,300
And first question, what is a
steady state or critical point?

10
00:00:41,300 --> 00:00:43,430
Easy question.

11
00:00:43,430 --> 00:00:47,020
I'm looking at
special points capital

12
00:00:47,020 --> 00:00:54,380
Y, where the right-hand side
is 0, special points where

13
00:00:54,380 --> 00:00:57,340
the function is 0.

14
00:00:57,340 --> 00:01:04,000
And I'll call those critical
points or steady states.

15
00:01:04,000 --> 00:01:04,930
What's the point?

16
00:01:04,930 --> 00:01:10,020
At a critical point,
here is the solution.

17
00:01:10,020 --> 00:01:10,950
It's a constant.

18
00:01:10,950 --> 00:01:11,515
It's steady.

19
00:01:15,520 --> 00:01:19,430
I'm just checking here that
the equation is satisfied.

20
00:01:19,430 --> 00:01:23,950
The derivative is 0
because it's constant,

21
00:01:23,950 --> 00:01:27,930
and f is 0 because
it's a critical point.

22
00:01:27,930 --> 00:01:29,640
So I have 0 equals 0.

23
00:01:29,640 --> 00:01:33,170
The differential equation
is perfectly good.

24
00:01:33,170 --> 00:01:39,970
So if I start at a critical
point, I stay there.

25
00:01:39,970 --> 00:01:42,770
That's not our central question.

26
00:01:42,770 --> 00:01:46,950
Our key question is, if
I start at other points,

27
00:01:46,950 --> 00:01:52,685
do I approach a critical
point, or do I go away from it?

28
00:01:52,685 --> 00:01:56,320
Is the critical point
stable and attractive,

29
00:01:56,320 --> 00:02:00,840
or is it unstable and repulsive?

30
00:02:00,840 --> 00:02:04,280
So the way to
answer that question

31
00:02:04,280 --> 00:02:07,530
is to look at the
equation when you're

32
00:02:07,530 --> 00:02:10,199
very near the critical point.

33
00:02:10,199 --> 00:02:14,350
Very near the critical point, we
could make the equation linear.

34
00:02:14,350 --> 00:02:18,680
We can linearize the equation,
and that's the whole trick.

35
00:02:18,680 --> 00:02:21,870
And I've spoken before,
and I'll do it again now

36
00:02:21,870 --> 00:02:23,530
for one equation.

37
00:02:23,530 --> 00:02:27,240
But the real message,
the real content

38
00:02:27,240 --> 00:02:32,010
comes with two or
three equations.

39
00:02:32,010 --> 00:02:34,780
That's what we see
in nature very often,

40
00:02:34,780 --> 00:02:40,990
and we want to know,
is the problem stable?

41
00:02:40,990 --> 00:02:41,490
OK.

42
00:02:41,490 --> 00:02:45,250
So what does linearize mean?

43
00:02:45,250 --> 00:02:47,740
Every function is
linear if you look at it

44
00:02:47,740 --> 00:02:49,516
through a microscope.

45
00:02:54,570 --> 00:02:58,980
Maybe I should say if you
blow it up near y equal Y,

46
00:02:58,980 --> 00:03:00,885
every function is linear.

47
00:03:00,885 --> 00:03:02,015
Here is f of y.

48
00:03:06,040 --> 00:03:08,320
Here it's coming
through-- it's a graph

49
00:03:08,320 --> 00:03:10,622
of f of y, whatever it is.

50
00:03:10,622 --> 00:03:14,640
If this we recognize
as the point capital Y,

51
00:03:14,640 --> 00:03:17,410
right, that's where
the function is 0.

52
00:03:17,410 --> 00:03:24,900
And near that point, my function
is almost a straight line.

53
00:03:24,900 --> 00:03:30,500
And the slope of that
tangent is the coefficient,

54
00:03:30,500 --> 00:03:33,630
and everything depends on that.

55
00:03:33,630 --> 00:03:37,440
Everything depends on whether
the slope is going up like

56
00:03:37,440 --> 00:03:43,650
that-- probably that's going to
be unstable-- or coming down.

57
00:03:43,650 --> 00:03:46,330
If it were coming
down, then the slope

58
00:03:46,330 --> 00:03:49,240
would be negative at
the critical point,

59
00:03:49,240 --> 00:03:51,360
and probably that
will be stable.

60
00:03:51,360 --> 00:03:52,040
OK.

61
00:03:52,040 --> 00:03:55,410
So I just have to do
a little calculus.

62
00:03:55,410 --> 00:04:00,260
The whole idea of linearizing
is the central idea of calculus.

63
00:04:00,260 --> 00:04:04,750
That we have curves,
but near a point,

64
00:04:04,750 --> 00:04:10,690
we can pretend-- they are
essentially straight if we

65
00:04:10,690 --> 00:04:12,710
focus in, if we zoom in.

66
00:04:12,710 --> 00:04:16,390
So this is a zooming-in
problem, linearization.

67
00:04:16,390 --> 00:04:16,980
OK.

68
00:04:16,980 --> 00:04:25,850
So if I zoom in the
function at some y.

69
00:04:25,850 --> 00:04:30,850
I'm zooming in around
the point capital Y.

70
00:04:30,850 --> 00:04:33,710
But you remember
the tangent line

71
00:04:33,710 --> 00:04:40,910
stuff is the function
at Y. So little y is

72
00:04:40,910 --> 00:04:43,180
some point close by.

73
00:04:43,180 --> 00:04:46,610
Capital Y is the crossing point.

74
00:04:46,610 --> 00:04:56,870
And this is the y minus Y times
the slope-- that's the slope--

75
00:04:56,870 --> 00:05:06,720
the slope at the critical point
there is all that's-- you see

76
00:05:06,720 --> 00:05:09,860
that the right-hand
side is linear.

77
00:05:09,860 --> 00:05:12,110
And actually, f of Y is 0.

78
00:05:12,110 --> 00:05:15,100
That's the point.

79
00:05:15,100 --> 00:05:18,810
So that I have just a
linear approximation

80
00:05:18,810 --> 00:05:22,180
with that slope and
a simple function.

81
00:05:22,180 --> 00:05:23,490
OK.

82
00:05:23,490 --> 00:05:27,590
So I'll use this approximation.

83
00:05:27,590 --> 00:05:31,370
I'll put that into the
equation, and then I'll

84
00:05:31,370 --> 00:05:34,620
have a linear equation,
which I can easily solve.

85
00:05:34,620 --> 00:05:35,760
Can I do that?

86
00:05:35,760 --> 00:05:41,090
So my plan is, take my
differential equation,

87
00:05:41,090 --> 00:05:43,690
look, focus near
the steady state,

88
00:05:43,690 --> 00:05:48,060
near the critical point
capital Y. Near that point,

89
00:05:48,060 --> 00:05:55,570
this is my good approximation
to f, and I'll just use it.

90
00:05:55,570 --> 00:05:58,130
So I plan to use
that right away.

91
00:05:58,130 --> 00:05:59,580
So now here's the linearized.

92
00:06:05,820 --> 00:06:13,790
So d by dt of y equals f of y.

93
00:06:13,790 --> 00:06:20,630
But I'm going to do
approximately equals this y

94
00:06:20,630 --> 00:06:26,530
minus capital Y times the slope.

95
00:06:26,530 --> 00:06:31,220
So the slope is my
coefficient little a

96
00:06:31,220 --> 00:06:34,045
in my first-order
linear equation.

97
00:06:34,045 --> 00:06:40,610
So I'm going back to chapter
1 for this linearization

98
00:06:40,610 --> 00:06:42,530
for one equation.

99
00:06:42,530 --> 00:06:48,770
But then the next
video is the real thing

100
00:06:48,770 --> 00:06:52,640
by allowing two equations
or even three equations.

101
00:06:52,640 --> 00:06:54,570
So we'll make a
small start on that,

102
00:06:54,570 --> 00:06:56,530
but it's really the next video.

103
00:06:56,530 --> 00:06:57,260
OK.

104
00:06:57,260 --> 00:06:58,860
So that's the equation.

105
00:06:58,860 --> 00:07:02,620
Now, notice that
I could put dy dt

106
00:07:02,620 --> 00:07:05,710
as-- the derivative
of that constant

107
00:07:05,710 --> 00:07:08,150
is 0, so I could
safely put it there.

108
00:07:08,150 --> 00:07:10,085
So what does this tell me?

109
00:07:10,085 --> 00:07:12,070
Let me call that number a.

110
00:07:14,980 --> 00:07:20,300
So I can solve that
equation, and the solution

111
00:07:20,300 --> 00:07:24,740
will be y minus capital
Y. It's just linear.

112
00:07:24,740 --> 00:07:28,070
The derivative is the
thing itself times a.

113
00:07:28,070 --> 00:07:34,850
It's the pure model of steady
growth or steady decay.

114
00:07:34,850 --> 00:07:43,810
y minus Y is, let's
say, some e to the at.

115
00:07:43,810 --> 00:07:45,200
Right?

116
00:07:45,200 --> 00:07:50,610
When I have a coefficient
in the linear equation ay,

117
00:07:50,610 --> 00:07:52,870
I see it in the exponential.

118
00:07:52,870 --> 00:07:57,840
So a less than 0 is stable.

119
00:08:03,840 --> 00:08:08,840
Because a less than
0, that's negative,

120
00:08:08,840 --> 00:08:12,210
and the exponential drops to 0.

121
00:08:12,210 --> 00:08:15,430
And that tells me that
y approaches capital

122
00:08:15,430 --> 00:08:20,200
Y. It goes to the critical
point, to the steady state,

123
00:08:20,200 --> 00:08:21,110
and not away.

124
00:08:21,110 --> 00:08:22,520
Example, example.

125
00:08:22,520 --> 00:08:24,880
Let me just take an
example that you've

126
00:08:24,880 --> 00:08:33,640
seen before, the logistic
equation, where the right side

127
00:08:33,640 --> 00:08:38,020
is, say, 3y minus y squared.

128
00:08:38,020 --> 00:08:38,520
OK.

129
00:08:38,520 --> 00:08:40,659
Not linear.

130
00:08:40,659 --> 00:08:44,470
So I plan to linearize after
I find the critical points.

131
00:08:44,470 --> 00:08:47,450
Critical points, this is 0.

132
00:08:47,450 --> 00:08:49,960
That equals 0 at--
I guess there will

133
00:08:49,960 --> 00:08:56,930
be two critical points because
I have a second-degree equation.

134
00:08:56,930 --> 00:09:05,900
When that is 0, it could be 0
at y equals 0 or at y equals 3.

135
00:09:05,900 --> 00:09:09,340
So two critical points,
and each critical point

136
00:09:09,340 --> 00:09:15,040
has its own linearization, its
slope at that critical point.

137
00:09:15,040 --> 00:09:20,670
So you see, if I
graph f of y here,

138
00:09:20,670 --> 00:09:29,700
this 3y minus y squared has--
there is 3y minus y squared.

139
00:09:29,700 --> 00:09:32,090
There is one critical point, 0.

140
00:09:32,090 --> 00:09:34,550
There is the other
critical point at 3.

141
00:09:34,550 --> 00:09:38,240
Here the slope is
positive-- unstable.

142
00:09:38,240 --> 00:09:41,030
Here the slope is
negative-- stable.

143
00:09:41,030 --> 00:09:45,060
So this is stable, unstable.

144
00:09:45,060 --> 00:09:49,730
And let me just push
through the numbers here.

145
00:09:52,260 --> 00:09:57,790
So the df dy, that's the slope.

146
00:09:57,790 --> 00:10:00,440
So I have to take the
derivative of that.

147
00:10:00,440 --> 00:10:02,830
Notice this is not my
differential equation.

148
00:10:02,830 --> 00:10:04,880
There is my
differential equation.

149
00:10:04,880 --> 00:10:09,530
Here is my linearization
step, my computation

150
00:10:09,530 --> 00:10:12,390
of the derivative, the slope.

151
00:10:12,390 --> 00:10:17,570
So the derivative of
that is 3 minus 2y,

152
00:10:17,570 --> 00:10:20,690
and I've got two
critical points.

153
00:10:20,690 --> 00:10:24,180
At capital Y equal 0, that's 3.

154
00:10:24,180 --> 00:10:29,580
And at capital Y equals 3,
it's 3 minus 6, it's minus 3.

155
00:10:29,580 --> 00:10:32,340
Those are the slopes
we saw on the picture.

156
00:10:32,340 --> 00:10:35,500
Slope up, the
parabola is going up.

157
00:10:35,500 --> 00:10:36,700
Slope down.

158
00:10:36,700 --> 00:10:39,093
So this will
correspond to unstable.

159
00:10:43,165 --> 00:10:46,450
So what does it mean
for this to be unstable?

160
00:10:46,450 --> 00:10:52,000
It means that the solution
Y equals 0, constant 0,

161
00:10:52,000 --> 00:10:55,020
solves the equation, no problem.

162
00:10:55,020 --> 00:10:59,590
If Y stays at 0, it's a
perfectly OK solution.

163
00:10:59,590 --> 00:11:00,710
The derivative is 0.

164
00:11:00,710 --> 00:11:02,670
Everything's 0.

165
00:11:02,670 --> 00:11:04,990
But if I move a
little away from 0,

166
00:11:04,990 --> 00:11:10,080
if I move a little way
from 0, then the 3y minus y

167
00:11:10,080 --> 00:11:12,700
squared, what does it look like?

168
00:11:12,700 --> 00:11:15,320
If I'm moving just
a little away from Y

169
00:11:15,320 --> 00:11:18,900
equals 0, away from
this unstable point,

170
00:11:18,900 --> 00:11:22,420
y squared will be
extremely small.

171
00:11:22,420 --> 00:11:23,690
So it's really 3y.

172
00:11:26,860 --> 00:11:29,920
The y squared will be
small near Y equals 0.

173
00:11:29,920 --> 00:11:31,040
Forget that.

174
00:11:31,040 --> 00:11:33,635
We have exponential
growth, e to the 3t.

175
00:11:38,930 --> 00:11:43,640
We leave the 0 steady
state, and we move on.

176
00:11:43,640 --> 00:11:46,530
Now, eventually
we'll move somewhere

177
00:11:46,530 --> 00:11:50,485
near the other steady state.

178
00:11:50,485 --> 00:11:57,510
At capital Y equals 3, the
slope of this thing is minus 3,

179
00:11:57,510 --> 00:12:02,090
and the negative one
will be the stable point.

180
00:12:02,090 --> 00:12:14,000
So where y minus 3, the
distance to the steady state,

181
00:12:14,000 --> 00:12:19,620
the critical point will grow
like e to the mi-- well,

182
00:12:19,620 --> 00:12:23,380
will decay, sorry, I said grow,
I meant decay-- will decay

183
00:12:23,380 --> 00:12:29,040
like e to the minus 3t because
the minus 3 in the slope

184
00:12:29,040 --> 00:12:31,180
is the minus 3 in the exponent.

185
00:12:34,070 --> 00:12:34,710
OK.

186
00:12:34,710 --> 00:12:38,270
That's not rocket
science, although it's

187
00:12:38,270 --> 00:12:41,440
pretty important for rockets.

188
00:12:41,440 --> 00:12:43,570
Let me just say
what's coming next

189
00:12:43,570 --> 00:12:47,590
and then do it in
the follow-up video.

190
00:12:47,590 --> 00:12:59,960
So what's coming next will be
two equations, dy dt and dz dt.

191
00:13:03,120 --> 00:13:04,910
I have two things.

192
00:13:04,910 --> 00:13:06,990
y and z, they depend
on each other.

193
00:13:06,990 --> 00:13:16,440
So the growth or decay of y
is given by some function f,

194
00:13:16,440 --> 00:13:24,330
and this is given by some
different function g, so

195
00:13:24,330 --> 00:13:25,050
f and g.

196
00:13:25,050 --> 00:13:29,180
Now, when do I
have steady state?

197
00:13:29,180 --> 00:13:30,480
When this is 0.

198
00:13:30,480 --> 00:13:31,520
When they're both 0.

199
00:13:31,520 --> 00:13:33,180
They both have to be 0.

200
00:13:33,180 --> 00:13:36,860
And then dy dt is
0, so y is steady.

201
00:13:36,860 --> 00:13:39,540
dz dt is 0, so z is steady.

202
00:13:39,540 --> 00:13:44,290
So I'm looking for-- I've
got two numbers to look for.

203
00:13:44,290 --> 00:13:48,150
And I've got two
equations, f of y-- oh,

204
00:13:48,150 --> 00:13:52,370
let me call that capital
Y, capital Z-- so those

205
00:13:52,370 --> 00:13:54,740
are numbers now-- equals 0.

206
00:13:54,740 --> 00:14:03,200
So I want to solve-- equals 0,
and g of capital Y, capital Z

207
00:14:03,200 --> 00:14:05,630
equals 0.

208
00:14:05,630 --> 00:14:07,170
Yeah, yeah.

209
00:14:07,170 --> 00:14:14,000
So both right-hand
sides should be 0,

210
00:14:14,000 --> 00:14:17,080
and then I'm in a steady state.

211
00:14:17,080 --> 00:14:21,360
But this is going to be like
more interesting to linearize.

212
00:14:21,360 --> 00:14:25,210
That's really the next video,
is how do you linearize?

213
00:14:25,210 --> 00:14:27,560
What does the
linearized thing look

214
00:14:27,560 --> 00:14:32,090
like when you have two functions
depending on two variables

215
00:14:32,090 --> 00:14:33,490
Y and Z?

216
00:14:33,490 --> 00:14:39,080
You're going to have, we'll
see, [? for ?] slopes-- well,

217
00:14:39,080 --> 00:14:40,300
you'll see it.

218
00:14:40,300 --> 00:14:43,340
So this is what's coming.

219
00:14:43,340 --> 00:14:46,970
And we end up with
a two-by-two matrix

220
00:14:46,970 --> 00:14:52,990
because we have two equations,
two unknowns, and a little more

221
00:14:52,990 --> 00:14:58,960
excitement than the
classical single equation,

222
00:14:58,960 --> 00:15:00,920
like a logistic equation.

223
00:15:00,920 --> 00:15:01,420
OK.

224
00:15:01,420 --> 00:15:03,650
Onward to two.