1
00:00:00,499 --> 00:00:01,460
GILBERT STRANG: OK.
2
00:00:01,460 --> 00:00:05,660
So this is the second
lecture about these pictures,
3
00:00:05,660 --> 00:00:09,990
in the phase plane
that's with axes y and y
4
00:00:09,990 --> 00:00:14,330
prime, for a second order
constant coefficient
5
00:00:14,330 --> 00:00:18,470
linear, good problem.
6
00:00:18,470 --> 00:00:19,160
Good problem.
7
00:00:19,160 --> 00:00:23,330
And you remember that
we study that equation
8
00:00:23,330 --> 00:00:31,360
by looking for special
solutions y equals e to the st.
9
00:00:31,360 --> 00:00:33,400
When we plug that
into the equation,
10
00:00:33,400 --> 00:00:37,250
we get this simple
quadratic equation.
11
00:00:37,250 --> 00:00:39,530
And everything depends on that.
12
00:00:39,530 --> 00:00:44,260
So today this video
is about the case
13
00:00:44,260 --> 00:00:47,340
when the roots are complex.
14
00:00:47,340 --> 00:00:51,680
You remember, so the
roots, complex roots, you
15
00:00:51,680 --> 00:00:56,080
have a real part, plus or
minus an imaginary part.
16
00:00:56,080 --> 00:01:01,720
And this happens when b
squared is smaller than 4ac.
17
00:01:01,720 --> 00:01:03,830
Because you remember,
there's a square root
18
00:01:03,830 --> 00:01:08,280
in the formula for the solution
of a quadratic equation.
19
00:01:08,280 --> 00:01:12,620
There's a square root
of b squared minus 4ac,
20
00:01:12,620 --> 00:01:16,770
the usual formula from school.
21
00:01:16,770 --> 00:01:21,420
And if b squared is smaller,
we have a negative number
22
00:01:21,420 --> 00:01:22,990
under the square root.
23
00:01:22,990 --> 00:01:25,560
And we get complex roots.
24
00:01:25,560 --> 00:01:28,370
So last time the
roots were real.
25
00:01:28,370 --> 00:01:32,420
The pictures in the phase
plane set off to infinity,
26
00:01:32,420 --> 00:01:38,990
or came in to 0, more or less
almost on straight lines.
27
00:01:38,990 --> 00:01:42,970
Now we're going to have
curves and spirals, because
28
00:01:42,970 --> 00:01:45,070
of the complex part.
29
00:01:45,070 --> 00:01:48,390
So here are the three
possibilities now.
30
00:01:48,390 --> 00:01:50,130
We had three last time.
31
00:01:50,130 --> 00:01:53,560
Here are the other three
with complex roots.
32
00:01:53,560 --> 00:01:57,050
So the complex, the
real part, everything
33
00:01:57,050 --> 00:02:03,230
depends on this real
part that the stability
34
00:02:03,230 --> 00:02:06,850
going in, going out,
staying on a circle
35
00:02:06,850 --> 00:02:09,900
depends on that real part.
36
00:02:09,900 --> 00:02:13,640
If the real part is
positive, then we go out.
37
00:02:13,640 --> 00:02:21,260
We have an exponential e
to the a plus i omega t.
38
00:02:23,970 --> 00:02:30,190
And if a is positive that e to
the at would blow up, unstable.
39
00:02:30,190 --> 00:02:32,170
So that's unstable.
40
00:02:32,170 --> 00:02:33,390
Here is a center.
41
00:02:33,390 --> 00:02:38,020
When a is zero, then we just
have e to the i omega t.
42
00:02:38,020 --> 00:02:39,660
That's the nicest example.
43
00:02:39,660 --> 00:02:41,430
I do that one first.
44
00:02:41,430 --> 00:02:44,075
So in that case we're just
going around in a circle
45
00:02:44,075 --> 00:02:46,620
or around in an ellipse.
46
00:02:46,620 --> 00:02:50,600
And finally, the
physical problem
47
00:02:50,600 --> 00:02:54,360
where we have damping,
but not too much damping.
48
00:02:54,360 --> 00:02:56,970
So the roots are still complex.
49
00:02:56,970 --> 00:02:58,630
But they're going in.
50
00:02:58,630 --> 00:03:04,930
Because if a is negative,
e to the at is going to 0.
51
00:03:04,930 --> 00:03:08,210
So that's a stable case.
52
00:03:08,210 --> 00:03:11,970
That's a physical case.
53
00:03:11,970 --> 00:03:16,340
We hope to have a little damping
in our system, and be stable.
54
00:03:16,340 --> 00:03:19,730
This one we could
say neutrally stable.
55
00:03:19,730 --> 00:03:22,010
This one is certainly unstable.
56
00:03:22,010 --> 00:03:27,360
Let me start with that,
the neutrally stable.
57
00:03:27,360 --> 00:03:31,920
Because that's the most famous
equation in second order
58
00:03:31,920 --> 00:03:34,240
equation in mechanics.
59
00:03:34,240 --> 00:03:38,170
It's pure oscillation, a
spring going up and down,
60
00:03:38,170 --> 00:03:43,440
an LC circuit going back
and forth, pure oscillation.
61
00:03:43,440 --> 00:03:46,720
And we see the solutions.
62
00:03:46,720 --> 00:03:53,160
So I've written-- I've taken
this particular equation.
63
00:03:53,160 --> 00:03:54,740
You notice no damping.
64
00:03:54,740 --> 00:03:56,630
There's no y prime term.
65
00:03:56,630 --> 00:03:57,320
OK.
66
00:03:57,320 --> 00:04:01,500
So here is the solution,
famous, famous solution.
67
00:04:01,500 --> 00:04:05,910
And y prime it
will be c1, I guess
68
00:04:05,910 --> 00:04:09,990
the derivative of the
cosine is minus omega times
69
00:04:09,990 --> 00:04:14,380
sine omega t, plus c2.
70
00:04:14,380 --> 00:04:20,600
The derivative of that
is omega cos omega t.
71
00:04:20,600 --> 00:04:25,630
So that's the y and y prime.
72
00:04:25,630 --> 00:04:30,170
So for every t, it's going
to be an easy figure.
73
00:04:30,170 --> 00:04:32,530
Here is y.
74
00:04:32,530 --> 00:04:34,400
Here is y prime.
75
00:04:34,400 --> 00:04:37,650
And that's the phase
plane, phase plane.
76
00:04:37,650 --> 00:04:42,810
So at each time t, I
have a y and a y prime.
77
00:04:42,810 --> 00:04:44,245
And it gives me a point.
78
00:04:44,245 --> 00:04:48,110
So let me put it in there.
79
00:04:48,110 --> 00:04:52,440
As time moves on
that point moves.
80
00:04:52,440 --> 00:04:55,080
And it's the picture
in the phase plane,
81
00:04:55,080 --> 00:04:57,320
the orbit sometimes
you could say,
82
00:04:57,320 --> 00:05:00,840
it's kind of like
a planet or a moon.
83
00:05:00,840 --> 00:05:05,740
So for that, what is
the orbit for that one?
84
00:05:05,740 --> 00:05:08,390
Well it goes around
in an ellipse.
85
00:05:08,390 --> 00:05:11,280
It would be a circle
with-- let me draw it.
86
00:05:13,810 --> 00:05:16,800
This is the case omega equal 1.
87
00:05:16,800 --> 00:05:20,150
In that case, in that
most famous case,
88
00:05:20,150 --> 00:05:24,310
we simply go around a circle.
89
00:05:24,310 --> 00:05:25,250
There's y.
90
00:05:25,250 --> 00:05:27,120
There's y prime.
91
00:05:27,120 --> 00:05:31,630
We have cosine and sine and
cos squared plus sine squared
92
00:05:31,630 --> 00:05:33,100
is 1 squared.
93
00:05:33,100 --> 00:05:37,780
And we're going around a circle
of radius 1, or another circle
94
00:05:37,780 --> 00:05:39,930
depending on the
initial condition.
95
00:05:39,930 --> 00:05:46,460
Here there's a factor
omega, giving an extra push
96
00:05:46,460 --> 00:05:48,050
to y prime.
97
00:05:48,050 --> 00:05:53,160
So if omega was 2, for
example, then we'd have a 2
98
00:05:53,160 --> 00:05:57,160
in y prime from the omega,
which is not in the y.
99
00:05:57,160 --> 00:05:59,860
And that would make y
prime a little larger.
100
00:05:59,860 --> 00:06:04,750
And it would be twice as--
it would go up to twice as--
101
00:06:04,750 --> 00:06:10,850
that's meant to be, meant to
be an ellipse with height 2 up
102
00:06:10,850 --> 00:06:17,320
there, or in general
omega, and 1 there.
103
00:06:17,320 --> 00:06:21,180
So in the y direction
there is no factor omega.
104
00:06:21,180 --> 00:06:23,670
And we just have
cosine and sine.
105
00:06:23,670 --> 00:06:28,260
And that would be a typical
picture in the phase plane.
106
00:06:28,260 --> 00:06:31,730
But if we started with
smaller initial conditions,
107
00:06:31,730 --> 00:06:35,950
we would travel on
another ellipse.
108
00:06:35,950 --> 00:06:38,200
But the point is--
and these are called,
109
00:06:38,200 --> 00:06:40,060
this picture is called a center.
110
00:06:44,210 --> 00:06:47,010
So that's one of the six
possibilities, and in some way,
111
00:06:47,010 --> 00:06:48,980
kind of the most beautiful.
112
00:06:48,980 --> 00:06:53,200
You get ellipsis
in the phase plane.
113
00:06:53,200 --> 00:06:54,370
They close off.
114
00:06:54,370 --> 00:06:59,870
Because the solution just
repeats itself every period.
115
00:06:59,870 --> 00:07:01,470
It's periodic.
116
00:07:01,470 --> 00:07:02,700
y is periodic.
117
00:07:02,700 --> 00:07:04,280
y prime is periodic.
118
00:07:04,280 --> 00:07:07,280
They come around again
and again and again.
119
00:07:07,280 --> 00:07:12,450
No energy is lost, conservation
of energy, perfection.
120
00:07:12,450 --> 00:07:18,735
And I would say neutrally
stable, neutral stability.
121
00:07:23,980 --> 00:07:26,670
The solution doesn't go into 0.
122
00:07:26,670 --> 00:07:28,260
Because there's no damping.
123
00:07:28,260 --> 00:07:30,500
It doesn't go out to infinity.
124
00:07:30,500 --> 00:07:32,030
Because there's constant energy.
125
00:07:32,030 --> 00:07:34,560
And that's the picture
in the phase plane.
126
00:07:34,560 --> 00:07:35,490
OK.
127
00:07:35,490 --> 00:07:36,820
So that's the center.
128
00:07:36,820 --> 00:07:41,950
And now I'll draw one
with a source, or a sink.
129
00:07:41,950 --> 00:07:44,730
I just have to change
the sign on damping
130
00:07:44,730 --> 00:07:46,640
to get source or sink.
131
00:07:46,640 --> 00:07:48,560
So let me do that.
132
00:07:48,560 --> 00:07:56,235
So now I'm going to do
a spiral source or sink.
133
00:07:59,590 --> 00:08:03,090
This is the unstable one,
going out to infinity.
134
00:08:03,090 --> 00:08:05,970
This is the stable
one coming in to 0.
135
00:08:05,970 --> 00:08:14,960
And let me do y double prime,
plus or maybe minus 4y prime,
136
00:08:14,960 --> 00:08:17,470
plus 4y equals 0.
137
00:08:17,470 --> 00:08:19,850
Suppose I take that equation.
138
00:08:19,850 --> 00:08:25,570
Then I have s squared
plus 4s, oh maybe--
139
00:08:25,570 --> 00:08:27,740
maybe 2 is a nicer number.
140
00:08:27,740 --> 00:08:30,340
2 is nicer than 4.
141
00:08:30,340 --> 00:08:35,510
Let me change this
to a 2, and a 2.
142
00:08:35,510 --> 00:08:41,210
And so I have s
squared plus 2s plus 2
143
00:08:41,210 --> 00:08:44,390
or minus 2s plus 2 equals 0.
144
00:08:44,390 --> 00:08:53,400
So those are my-- positive
damping would be with a plus.
145
00:08:53,400 --> 00:09:01,010
So with a plus sign, the roots
are s squared plus 2s plus 2.
146
00:09:01,010 --> 00:09:08,200
The roots are 1, or rather
a minus 1 plus or minus i.
147
00:09:12,270 --> 00:09:19,890
Plus sign, and then the
minus sign, with a minus 2.
148
00:09:19,890 --> 00:09:26,320
Then all the roots have
a plus, plus or minus i.
149
00:09:26,320 --> 00:09:30,890
Everything is depending on
these roots, these exponents,
150
00:09:30,890 --> 00:09:34,900
which are the solutions of
the special characteristic
151
00:09:34,900 --> 00:09:37,970
equation, the simple
quadratic equation.
152
00:09:37,970 --> 00:09:42,190
And you see that depending
on positive damping
153
00:09:42,190 --> 00:09:48,110
or a negative damping, I get
stability or instability.
154
00:09:48,110 --> 00:09:51,240
And let me draw a picture.
155
00:09:51,240 --> 00:09:54,050
I don't if I can try two
pictures in the same thing,
156
00:09:54,050 --> 00:09:54,830
probably not.
157
00:09:54,830 --> 00:09:56,170
That wouldn't be smart.
158
00:09:56,170 --> 00:09:58,620
So what's happening then?
159
00:09:58,620 --> 00:10:00,040
Let's take this one.
160
00:10:00,040 --> 00:10:06,010
So this solution y
is e to the minus t.
161
00:10:06,010 --> 00:10:10,120
That's what's making it
stable coming into 0, times--
162
00:10:10,120 --> 00:10:20,810
and from here we have
c1 cos t and c2 sine t.
163
00:10:20,810 --> 00:10:23,460
That's what we get
from the usual,
164
00:10:23,460 --> 00:10:28,560
as in the case of a center that
carries us around the circle.
165
00:10:28,560 --> 00:10:32,540
So what's happening in this
picture, in this phase plane?
166
00:10:32,540 --> 00:10:38,080
Here's a phase plane
again, y and y prime.
167
00:10:38,080 --> 00:10:41,270
Without the minus
1, we have a center.
168
00:10:41,270 --> 00:10:43,830
We just go around in a circle.
169
00:10:43,830 --> 00:10:47,780
But now because of
the minus 1, which
170
00:10:47,780 --> 00:10:51,380
is the factor e to the
minus t in the solution,
171
00:10:51,380 --> 00:10:54,360
as we go around we come in.
172
00:10:54,360 --> 00:10:57,300
And the word for that
curve is a spiral.
173
00:10:57,300 --> 00:11:08,130
So this would be the center,
going around in a circle.
174
00:11:08,130 --> 00:11:10,630
But now suppose we start here.
175
00:11:10,630 --> 00:11:17,210
Suppose we start at y equal
1, and y prime equal 0,
176
00:11:17,210 --> 00:11:20,090
start there at time 0.
177
00:11:20,090 --> 00:11:21,190
Let time go.
178
00:11:21,190 --> 00:11:24,980
Plot where we go.
179
00:11:24,980 --> 00:11:27,310
Where does this y
and the y prime,
180
00:11:27,310 --> 00:11:29,770
where is the point, y, y prime?
181
00:11:29,770 --> 00:11:30,410
OK.
182
00:11:30,410 --> 00:11:35,950
I'm starting it at-- so I'm
probably taking c1 as 1,
183
00:11:35,950 --> 00:11:37,560
and c2 as 0.
184
00:11:37,560 --> 00:11:39,780
So I'm starting it right there.
185
00:11:39,780 --> 00:11:45,680
And then I'll travel
along, depending on sines.
186
00:11:45,680 --> 00:11:51,330
I would go, I think,
probably this way.
187
00:11:51,330 --> 00:11:55,150
So it will travel
on a-- it comes in
188
00:11:55,150 --> 00:11:56,960
pretty fast, of course.
189
00:11:56,960 --> 00:12:00,780
Because that exponential
is a powerful guy
190
00:12:00,780 --> 00:12:02,460
that e to the minus t.
191
00:12:02,460 --> 00:12:12,980
So this is the solution,
damping out to 0.
192
00:12:12,980 --> 00:12:17,890
That's with the plus
sign, plus damping,
193
00:12:17,890 --> 00:12:23,240
which gives the minus sign
in the s, in the exponent.
194
00:12:23,240 --> 00:12:26,310
And then so that
is a spiral sink.
195
00:12:29,670 --> 00:12:32,790
Sink meaning just as
water in a bathtub
196
00:12:32,790 --> 00:12:35,330
flows in, that's what happens.
197
00:12:35,330 --> 00:12:39,260
Now what happens in a
spiral source that's
198
00:12:39,260 --> 00:12:41,490
what we have with a minus sign.
199
00:12:41,490 --> 00:12:43,110
Now we have a 1.
200
00:12:43,110 --> 00:12:45,500
Now we have an e to the plus t.
201
00:12:45,500 --> 00:12:46,940
Everything is growing.
202
00:12:46,940 --> 00:12:53,340
So instead of decaying, we're
going around but growing-- OK.
203
00:12:53,340 --> 00:12:59,110
I'm off the board, way off
the board with that spiral.
204
00:12:59,110 --> 00:13:01,170
Which is going to
keep going around,
205
00:13:01,170 --> 00:13:05,240
but explode out to infinity.
206
00:13:05,240 --> 00:13:08,640
So those are the
three possibilities
207
00:13:08,640 --> 00:13:14,790
for complex roots, centers,
spiral source, and spiral sink.
208
00:13:14,790 --> 00:13:18,520
For real roots we
had ordinary source,
209
00:13:18,520 --> 00:13:21,070
and ordinary sink, no spiral.
210
00:13:21,070 --> 00:13:25,060
And then the other possibility
was a saddle point,
211
00:13:25,060 --> 00:13:28,000
where almost surely we go out.
212
00:13:28,000 --> 00:13:32,160
But there was one direction
that came into the saddle point.
213
00:13:32,160 --> 00:13:32,660
OK.
214
00:13:32,660 --> 00:13:35,560
Those six pictures
are going to control
215
00:13:35,560 --> 00:13:39,790
the whole problem of stability,
which is our next subject.
216
00:13:39,790 --> 00:13:41,570
Thank you.