1
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So we've been looking at period-doubling
2
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and here again in this picture
3
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we have period-doubling
4
00:00:09,268 --> 00:00:12,105
from two to four, four to eight, and so on.
5
00:00:12,105 --> 00:00:15,610
Capital delta is the range of parameter values
6
00:00:15,610 --> 00:00:19,343
for which we have period 2.
7
00:00:19,343 --> 00:00:21,380
Delta two is the next range,
8
00:00:21,380 --> 00:00:23,320
delta three is the next range,
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00:00:23,320 --> 00:00:25,860
and we define these ratios
10
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this divided by this,
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this divided by that,
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00:00:28,614 --> 00:00:31,407
to be these little deltas.
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00:00:31,407 --> 00:00:32,885
And for the logistic equation,
14
00:00:32,885 --> 00:00:35,927
we found that delta one is about 4.75,
15
00:00:35,927 --> 00:00:41,598
and delta two is about 4.65/4.66.
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00:00:42,643 --> 00:00:44,052
In the quiz that you just did,
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00:00:44,052 --> 00:00:47,906
you did the same thing for the cubic equation.
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00:00:47,906 --> 00:00:54,961
And you found 4.419 and 4.618.
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So the numbers for delta one and delta two
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00:01:00,195 --> 00:01:02,458
for these equations aren't similar--
21
00:01:02,458 --> 00:01:04,481
sorry, ARE similar, but not identical.
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00:01:04,481 --> 00:01:07,858
However, if we kept going on and calculate delta three
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00:01:07,858 --> 00:01:10,622
and delta four and delta five and so on
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these numbers would get closer and closer to each other.
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So let me write that,
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00:01:17,017 --> 00:01:19,561
we're interested in delta n--
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which is capital delta n over capital delta n plus one.
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And this is going to go to the number 4.669201.
29
00:01:37,055 --> 00:01:39,292
So for both of these equations,
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00:01:39,292 --> 00:01:40,988
as we let n get larger and larger,
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00:01:40,988 --> 00:01:43,174
we get closer and closer to this transition point,
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00:01:43,174 --> 00:01:48,319
these ratios approach 4.669201.
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00:01:48,319 --> 00:01:52,762
So in the large n limit, this number is just known as delta.
34
00:01:52,762 --> 00:02:00,810
And we would say that delta is universal.
35
00:02:02,162 --> 00:02:04,053
so delta is universal,
36
00:02:04,053 --> 00:02:05,867
and what that means is
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00:02:05,867 --> 00:02:07,472
that it has the same value for
38
00:02:07,472 --> 00:02:10,742
a very large family or class of functions.
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00:02:10,742 --> 00:02:15,164
Let me state this a little bit more carefully,
40
00:02:15,164 --> 00:02:17,543
since it's a crucial result.
41
00:02:17,543 --> 00:02:20,444
So again, the phenomenon of universality:
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00:02:20,444 --> 00:02:23,637
We have delta n approaching a number delta,
43
00:02:23,637 --> 00:02:25,744
and we say that delta is universal.
44
00:02:25,744 --> 00:02:27,004
And the result is
45
00:02:27,004 --> 00:02:29,347
that this number delta is the same,
46
00:02:29,347 --> 00:02:34,007
4.669201 for all iterated functions
47
00:02:34,007 --> 00:02:35,539
that map an interval to itself
48
00:02:35,539 --> 00:02:38,959
and have a single quadratic maximum.
49
00:02:38,959 --> 00:02:44,964
So let me say a little bit about these conditions.
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00:02:44,964 --> 00:02:47,899
So if the function maps an interval to itself,
51
00:02:47,899 --> 00:02:51,359
that just excludes the possibility that orbits go off to infinity,
52
00:02:51,359 --> 00:02:55,048
either positive or negative infinity.
53
00:02:55,048 --> 00:02:56,113
So the logistic equation is an example.
54
00:02:56,113 --> 00:02:57,715
It mapped the unit interval to itself,
55
00:02:57,715 --> 00:02:59,312
meaning numbers between 0 and 1
56
00:02:59,312 --> 00:03:01,263
remain between 0 and 1.
57
00:03:01,263 --> 00:03:03,796
Let me draw some pictures to illustrate this:
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00:03:03,796 --> 00:03:04,580
What does it mean
59
00:03:04,580 --> 00:03:08,880
for a function to have a single quadratic maximum?
60
00:03:08,880 --> 00:03:16,573
So single quadratic maximum...
61
00:03:29,356 --> 00:03:31,540
So our favorite example, the logistic equation,
62
00:03:31,540 --> 00:03:33,933
that's just an upside down parabola.
63
00:03:33,933 --> 00:03:35,986
It has a single maximum right here
64
00:03:35,986 --> 00:03:38,000
and it's quadratic because well,
65
00:03:38,000 --> 00:03:39,731
it's a parabola, its a quadratic function.
66
00:03:39,731 --> 00:03:46,549
Here's another example though.
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00:03:46,549 --> 00:03:49,775
This is let's see...
68
00:03:49,775 --> 00:03:53,754
the cubic function looks something like this.
69
00:03:53,754 --> 00:03:55,468
It's not symmetric
70
00:03:55,468 --> 00:03:58,228
and the function is not exactly a parabola.
71
00:03:58,228 --> 00:04:00,896
However, if we look closely near the maximum,
72
00:04:00,896 --> 00:04:04,442
it would become more and more parabola-like.
73
00:04:04,442 --> 00:04:06,376
So it's locally parabolic.
74
00:04:06,376 --> 00:04:08,860
And in calculus terms, this just means
75
00:04:08,860 --> 00:04:12,094
that the second derivative doesn't disappear.
76
00:04:12,094 --> 00:04:14,900
So most functions that have a peak like this,
77
00:04:14,900 --> 00:04:17,259
will behave this way.
78
00:04:17,259 --> 00:04:24,104
Let me give some counterexamples.
79
00:04:25,591 --> 00:04:28,663
This function does not have a single quadratic maximum
80
00:04:28,663 --> 00:04:30,167
because it does not have a single maximum--
81
00:04:30,167 --> 00:04:33,970
it has two: a maximum here and then there.
82
00:04:33,970 --> 00:04:36,766
So this function does not meet the criteria
83
00:04:36,766 --> 00:04:41,982
in that statement I gave on the previous piece of paper.
84
00:04:41,982 --> 00:04:44,440
A few more counterexamples:
85
00:04:44,440 --> 00:04:46,848
A function like this,
86
00:04:46,848 --> 00:04:51,423
this has a single maximum, a nice peak,
87
00:04:51,423 --> 00:04:53,800
but it's not a quadratic maximum.
88
00:04:53,800 --> 00:04:55,719
If you zoom in on this point,
89
00:04:55,719 --> 00:04:57,417
it doesn't start to look like a parabola,
90
00:04:57,417 --> 00:04:58,812
it keeps looking like a point.
91
00:04:58,812 --> 00:05:00,261
The idea is that this is infinitely-sharp.
92
00:05:00,261 --> 00:05:01,920
Here if you zoom in on this,
93
00:05:01,920 --> 00:05:06,823
you can make it look as close to a parabola as you wish.
94
00:05:06,823 --> 00:05:11,898
So this does not have a single quadratic maximum
95
00:05:11,898 --> 00:05:13,841
because, well, it has a maximum,
96
00:05:13,841 --> 00:05:18,285
but it's not quadratic--it's pointy!
97
00:05:18,285 --> 00:05:20,736
Here's one more example.
98
00:05:20,736 --> 00:05:24,569
It has a maximum, but it's flat,
99
00:05:24,569 --> 00:05:27,589
so it doesn't really have a single maximum value.
100
00:05:27,589 --> 00:05:29,480
And if you zoom in on this line,
101
00:05:29,480 --> 00:05:31,286
again, it won't look like a parabola,
102
00:05:31,286 --> 00:05:34,163
it will look like a line.
103
00:05:34,163 --> 00:05:36,946
So this is also not a function
104
00:05:36,946 --> 00:05:41,007
with a single quadratic maximum.
105
00:05:41,007 --> 00:05:43,989
So any function with a single quadratic maximum
106
00:05:43,989 --> 00:05:46,204
it could be a parabola, a cubic function,
107
00:05:46,204 --> 00:05:48,844
sine functions, various exponentials,
108
00:05:48,844 --> 00:05:52,382
there are many many examples of such functions.
109
00:05:52,382 --> 00:05:54,679
And this is a pretty generic criteria
110
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in that if you just draw a function by hand
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00:05:58,356 --> 00:06:00,630
or cook something up that has a single maximum,
112
00:06:00,630 --> 00:06:03,420
odds are it'll be smooth
113
00:06:03,420 --> 00:06:06,471
and this criteria will hold.
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00:06:06,471 --> 00:06:09,025
So this isn't a very restrictive criteria.
115
00:06:09,025 --> 00:06:13,932
And there's a vast number of functions that meet this criteria.
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00:06:13,932 --> 00:06:18,350
So let me state this once more
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00:06:18,350 --> 00:06:20,654
This property delta,
118
00:06:20,654 --> 00:06:22,590
which is a feature of how the
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00:06:22,590 --> 00:06:30,060
sideways used those shapes on the bifurcation diagram,
120
00:06:30,060 --> 00:06:32,209
how they're related to each other,
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how much smaller each one gets
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as we get closer to this transition,
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00:06:37,620 --> 00:06:40,543
that this geometric quantity delta is universal.
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It has the same value, 4.669201,
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00:06:43,699 --> 00:06:47,095
and it goes on and on for all iterated functions f(x)
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that map an interval to itself and
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have a single quadratic maximum.
128
00:06:51,140 --> 00:06:54,334
So again, this is not a very restrictive criteria.
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00:06:54,334 --> 00:06:56,579
So any function you come up with
130
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that meets this very mild criteria,
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00:06:58,632 --> 00:07:00,915
you can make a bifurcation diagram
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find the bifurcation points like we did,
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00:07:06,083 --> 00:07:07,607
calculate the deltas,
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00:07:07,607 --> 00:07:09,086
go deeper and deeper
135
00:07:09,086 --> 00:07:10,432
and let n get larger and larger,
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00:07:10,432 --> 00:07:12,044
and this number will appear.
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00:07:12,044 --> 00:07:20,852
So this number is a property of all of those functions.
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00:07:20,852 --> 00:07:22,507
So I want to emphasize
139
00:07:22,507 --> 00:07:25,544
just how amazing this result is.
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We started with the logistic equation,
141
00:07:28,554 --> 00:07:30,246
just about as simple a nonlinear equation
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00:07:30,246 --> 00:07:31,942
as one could imagine,
143
00:07:31,942 --> 00:07:34,679
and we saw that we had this bifurcation diagram
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with an amazing amount of complexity,
145
00:07:37,533 --> 00:07:39,520
but also with some regularity--
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00:07:39,520 --> 00:07:41,109
we saw those pitchforks,
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00:07:41,109 --> 00:07:42,771
those sideways Us,
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00:07:42,771 --> 00:07:44,625
repeating again and again and again,
149
00:07:44,625 --> 00:07:47,219
and we noticed that there's
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00:07:47,219 --> 00:07:49,261
sort of a geometric similarity to them,
151
00:07:49,261 --> 00:07:51,393
that the next pitchfork in the sequence
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is smaller than the previous one,
153
00:07:53,460 --> 00:07:56,534
but maybe smaller by the same factor.
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00:07:56,534 --> 00:08:00,442
So that led us to investigate that idea quantitatively.
155
00:08:00,442 --> 00:08:02,279
And we defined these lowercase deltas
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00:08:02,279 --> 00:08:07,157
as that ratio of one pitchfork length
157
00:08:07,157 --> 00:08:09,137
to the next pitchfork length.
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00:08:09,137 --> 00:08:13,943
And we found that that ratio approaches a constant number,
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00:08:13,943 --> 00:08:16,032
this number 4.669.
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00:08:16,032 --> 00:08:20,067
And that's an interesting result for the logistic equation.
161
00:08:20,067 --> 00:08:26,104
It's a practical mathematical oddities tells us about
162
00:08:26,104 --> 00:08:29,812
the geometry of this particular bifurcation diagram.
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Then we looked at bifurcation diagrams
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00:08:32,220 --> 00:08:34,617
for other equations--quite different equations:
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00:08:34,617 --> 00:08:37,152
sines and cubes as well as parabolas--
166
00:08:37,152 --> 00:08:42,117
and we saw the same general features of the bifurcation diagram,
167
00:08:42,117 --> 00:08:45,408
but then if we calculate these deltas,
168
00:08:45,408 --> 00:08:47,011
and we kept calculating them
169
00:08:47,011 --> 00:08:48,966
to larger and larger periods,
170
00:08:48,966 --> 00:08:54,271
we would find again the same number appearing--4.669.
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00:08:54,271 --> 00:08:56,600
So that's where things started to get really weird--
172
00:08:56,600 --> 00:09:01,075
that we're seeing the same number appearing
173
00:09:01,075 --> 00:09:03,960
in very different equations.
174
00:09:03,960 --> 00:09:04,719
So that tells us,
175
00:09:04,719 --> 00:09:06,080
at some level the equations,
176
00:09:06,080 --> 00:09:08,608
or the details of the equations don't matter.
177
00:09:08,608 --> 00:09:10,555
There's some broader or overarching
178
00:09:10,555 --> 00:09:12,349
feature of these systems thats
179
00:09:12,349 --> 00:09:14,430
independent of the particular equation
180
00:09:14,430 --> 00:09:15,731
that we use.
181
00:09:15,731 --> 00:09:16,929
So that's something deep and
182
00:09:16,929 --> 00:09:20,233
something pretty surprising I think.
183
00:09:20,233 --> 00:09:22,134
Let me mention just a tiny bit about
184
00:09:22,134 --> 00:09:23,826
the history of this idea and this realization.
185
00:09:23,826 --> 00:09:28,135
The results I presented date from 1978.
186
00:09:28,135 --> 00:09:30,631
They're usually attributed to Mitchell Feigenbaum,
187
00:09:30,631 --> 00:09:33,446
and American physicist who discovered
188
00:09:33,446 --> 00:09:35,854
this property and then did some
189
00:09:35,854 --> 00:09:41,158
analytical work to try to understand it further.
190
00:09:41,158 --> 00:09:43,315
But it was also discovered independently by
191
00:09:43,315 --> 00:09:47,363
Charles Tresser and Pierre Coule,
192
00:09:47,363 --> 00:09:48,228
around the same time,
193
00:09:48,228 --> 00:09:50,655
also published in 1978.
194
00:09:50,655 --> 00:09:52,670
So this is a relatively new result.
195
00:09:52,670 --> 00:09:55,329
Another thing I want to mention is that
196
00:09:55,329 --> 00:09:59,759
I presented these results experimentally--
197
00:09:59,759 --> 00:10:02,312
the result of doing some numerical work on a computer,
198
00:10:02,312 --> 00:10:04,561
but there's a lot of very elegant
199
00:10:04,561 --> 00:10:06,343
and very powerful analytic work
200
00:10:06,343 --> 00:10:09,730
that calculates these numbers--4.669201--
201
00:10:09,730 --> 00:10:13,636
and explains why this number appears again and again.
202
00:10:13,636 --> 00:10:15,232
The mathematical framework for
203
00:10:15,232 --> 00:10:17,219
carrying out that analysis is know as
204
00:10:17,219 --> 00:10:18,638
a renormalization group,
205
00:10:18,638 --> 00:10:20,311
or just renormalization,
206
00:10:20,311 --> 00:10:22,039
and I'll say a little bit about that
207
00:10:22,039 --> 00:10:23,594
in a subsequent unit.
208
00:10:23,594 --> 00:10:25,166
I can't explain it in detail.
209
00:10:25,166 --> 00:10:27,670
It's just too much math for the level of this course.
210
00:10:27,670 --> 00:10:31,789
But I can maybe give some sort of an argument for that.
211
00:10:31,789 --> 00:10:36,338
But before I do that, I want to mention the next thing,
212
00:10:36,338 --> 00:10:38,327
which is,
213
00:10:38,327 --> 00:10:40,514
that this result is not just mathematics,
214
00:10:40,514 --> 00:10:42,926
So it's an amazing mathematical result,
215
00:10:42,926 --> 00:10:44,506
these one-dimensional functions
216
00:10:44,506 --> 00:10:46,576
have these beautiful bifurcation diagrams
217
00:10:46,576 --> 00:10:47,868
with the same branching,
218
00:10:47,868 --> 00:10:50,419
or same sort of fork-ratio in them.
219
00:10:50,419 --> 00:10:52,268
But this is physics as well.
220
00:10:52,268 --> 00:10:55,891
Period-doubling occurs in real physical systems,
221
00:10:55,891 --> 00:10:58,133
and one can go out and measure the rates
222
00:10:58,133 --> 00:11:02,593
at which those period-doublings occur.
223
00:11:02,593 --> 00:11:06,155
And one finds again, this number 4.669.
224
00:11:06,155 --> 00:11:07,250
So I'll describe that result
225
00:11:07,250 --> 00:11:08,726
in the next lecture.