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## ← Nonlinear 1.1 Introduction to nonlinear dynamics

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Showing Revision 13 created 07/20/2016 by Steven Gunawan.

1. Hi this is Liz Bradley, I'm a Professor
in the Computer Science department
2. at the University of Colorado at Boulder
and also on the external faculty of the
3. Santa Fe Institute. My research interests
are in nonlinear dynamics and chaos and
4. in artificial intelligence, and I'm going
to be your guide during this course on
5. nonlinear dynamics and chaos. Here's an
example of a nonlinear dynamical system.
6. It's a double pendulum. Two pieces of
aluminium and four ball bearings. Even
7. though the system is physically very
simple, it's behavior is very complicated.
8. Moreover, this system is sensitively
dependent on initial conditions. If I
9. started here, or here, the future evolution
of the behavior will be very different.
10. Even though the behavior of that device is
very very complicated, there are some very
11. strong patterns in that behavior, and the
tandem of those patterns and the sensitivity
12. is the hallmark of chaos. Now there's
lots of words on this slide that we'll get
13. into over the next ten weeks. I'll just
give you some highlights here.
14. A deterministic system is one that is not
random. Cause and effect are linked and
15. the current state determines the future
state.
16. A dynamic system (or a dynamical system),
either are fine, is a system that evolves with time
17. A nonlinear system is one where the
relationships between the variables that
18. matter are not linear. An example of a non
linear system is the gas gauge in a car,
19. at least in my car, where I fill up the
tank, and then I drive a hundred miles and
20. the needle barely moves. And then I drive
another hundred miles and the needle
21. plummets. That's a nonlinear relationship
between the level of gas in the tank
22. and the position of the needle. Now non
linear dynamics and chaos are not rare.
23. Of all the systems in the universe that
evolve with time, that's the outer
24. ellipse in this Venn diagram, the vast
majority of them are nonlinear.
25. Indeed a famous mathematician refers to
the study of nonlinear dynamics as the
26. study of non-elephant animals. Now this is
somewhat problematic, because the
27. traditional training that we get in
science, engineering and mathematics uses
28. the assumption of linearity, and that's
only a very small part of the picture.
29. Now looking at the inner two ellipses on
this Venn diagram conveys the point that
30. the majority of nonlinear systems are
chaotic, and so that's gonna play a big
31. role in this course. And the equations
that describe chaotic systems cannot be
32. solved analytically, that is with paper
and pencil, rather we have to solve them
33. with computers. And that is a large part
of what distinguishes this course on
34. nonlinear dynamics and chaos from most
other courses on this topic area,
35. including Steve Strogatz's great lectures
which are on the web, and the courses on
topic. We will focus not only on the
37. mathematics, but also on the role of
computation in the field. In this field,
38. the computer is the lab instrument. This
is experimental mathematics. And that's
39. actually why the field of nonlinear dynamics
only took of three or four decades ago
40. Before that, there weren't computers to
help us solve the equations. Now to
41. succeed in this course, you'll need to
understand the notion of a derivative,
42. because dynamical systems are about change
with time, and derivatives are the
43. mathematics of change with time. You'll
also need to be able to write simple
44. computer programs. Basically, to translate
simple mathematics formulas into code, run
45. them, and plot the results, say on the
axis of x versus t. There is no required
46. computer language. You can use
whichever programming language you want.
47. And you're not gonna turn in your code in
this course. We're interested in the
48. results that come out of it. You'll also
need to know about basic classical
49. mechanics, the stuff that you get in first
semester physics, like pendulums and
50. masses on springs, and bodies pulling on
each other, with GmM over r-squared kinds
51. of forces. Speaking of GmM over r-squared,
you may have seen this movie in the promo
52. video that I made. This is movie taken by
a camera on the Cassidy spacecraft as it
53. flew by Saturn's moon, Hyperion. Hyperion
is a very unusual shape and as a result of
54. that shape, it tumbles chaotically.
There's also chaos on how planets move
55. through space, not just how they tumble.
You may remember from Physics, that the
56. solutions in those cases can only be conic
sections, ellipses, parabolas and
57. hyperbolas. As we will see, systems with
three or more bodies can be chaotic. Now
58. think about it, how many bodies are there
in the solar system: lots more than two.
59. Indeed several hundred years, the King of
Sweden issued the challenge of a large
60. cash prize to the person who could prove
whether or not the solar system was stable
61. in the long term, and that prize was never
claimed. But the answer appeared in the
62. 1980s. Indeed the solar system is chaotic,
although it is stable in a sense and we'll
63. get back to that. So just some brief
history of our field, it really dates back
64. to Henri Poincare in the late 1800s. But
it really got going in the 1960s with Ed
65. Lorentz's paper, called Deterministic Non
periodic Flow. Lorentz was the first
66. person to recognize the patterns of chaos
and the sensitivity of the evolution of
67. the system, within the context of those
patterns. In the 70s, this paper by Li and
68. Yorke was the first to use the word
"chaos" in conjunction with this behavior.
69. In the late 70s and 80s, the chaos cabal
at the University of California at Santa
70. Cruz, got very interested in nonlinear
dynamics, and one of the problems that
71. they approached it with was trying to beat
roulette, that is, modelling the path of a
72. ball on a roulette wheel, and using that
information to advantage. After this,
73. things really took off. And I should say,
of course, that I'm only cherry-picking a
74. very small number of examples by lots of
smart people in a very active field.
75. Nonlinear dynamics turns up all over the
place. Imagine an eddy in a creek, so a
76. patch of swirling water on the surface of
a creek or a river, you can imagine
77. dropping a wood chip in that patch of
water and watching its path from above,
78. perhaps with a camera, and then dropping
another wood chip in that eddy at a
79. slightly different point, and watching its
path. Those paths, they will trace out
80. the patches of swirling water in that eddy
in different order, but if you did a time
81. lapse photograph of their paths, they
would both trace out the same eddy.
82. Weather is nonlinear and chaotic. You may
have heard of the butterfly effect.
83. A butterfly flapping its wings setting off
a hurricane a week later, a thousand miles
84. away. Again, small change, large effect,
sensitive dependence on initial conditions
85. Marine invertebrates actually make use of
chaotic mixing in the water around them
86. during spawning, and I'm interested in
exploiting chaotic mixing to design better
87. fuel injectors in cars. Nonlinear and
chaotic dynamics also turns up in driven
88. nonlinear oscillators, like the pendulum
that I showed you, like the human heart
89. which is normally kind of mostly periodic
but, can go into a chaotic state called
90. ventricular fibrillation and as you saw
with the example of Hyperion, there's a
91. lot of nonlinear and chaotic dynamics in
classical mechanics ranging from the three
92. body problem to how black holes move
around each other. And nonlinear and
93. chaotic dynamics turns up in lots and lots
of other fields, including, certainly,
94. things that you are interested in. So as I
hope you can see, nonlinear and chaotic
95. dynamics are not an academic oddity. They
are widespread, and they are fascinating,
96. and I hope that you will get infected by
some of that fascination over the course
97. of the next ten weeks. There are other
fascinating courses on the Complexity
98. Explorer website including Dave Feldman's
course on the same topic area that only
99. assumes knowledge of high school algebra,
and Melanie Mitchell's wonderful course on
100. complexity. The difference between
complexity and chaos actually bears a
101. little bit of explanation. Put perhaps too
simply, you can think of chaos as
102. complicated behavior from simple systems,
like my pendulum. And you can think of
103. complexity science as addressing systems
that are very complicated but have simple
104. behavior. Again, that is too pat but the
idea is generally right. So, a thousand
105. fish forming a single school. Now, some
logistics. There are several thousand of
106. you and one of me. We have an email
address for this course but it can very
107. rapidly get overwhelmed. Please do not use
my own personal email address, or that of
108. the TA, for course-related communications.
That thousands-to-one ratios is one of the
109. major issues with MOOCs like this one.
Part of the way we plan to work around
110. that is with an electronic forum. This is
not just to take a load off the course
111. staff, it's also to solve one of the other
problems with MOOCs, which is, instead of
112. being in a traditional classroom, everyone
taking this course is working by themselves
113. all over the world in all sorts of time
zones. And we hope to use the forum to
114. help with that. So if you've a question,
look on the forum. Someone else may have
115. posted that question already. If not,
post it yourself. If someone has posted an
116. answer, look at that answer. If you see a
question that you know the answer to, or
117. you think you do, offer your answer. I'll
also use the forum, by the way, to post
118. announcements, like there's a bug in the
problem set, or I've just posted a whole
119. new unit, or, the New York Times has an
article about the stuff I just talked about.
120. I'll also post discussion questions and
answers for topics that may interest some
121. people in the course, if somebody wants to
go deeper into something or sideways along
122. a tangent, that's where the forum can
play a role.
123. Here's another piece of technology that
can help.
124. There's no textbooks for this course. I'm
pulling together material from many many
125. different sources, including a substantial
amount from my own work, papers that I've
126. read, talks that I've heard at conferences
and so on and so forth.
127. These video lectures are short, self-
contained summaries of each topic. I use
128. the Supplementary Materials page to
supplement those summaries. So if you want
129. to dig more deeply into something I
mentioned, or you'd like some background
130. material, or, you wanna read the original
paper that I mentioned. This is where you
131. should look. In the next segment of this
course, we'll start digging into some
132. ideas and mathematics and plots and
computer examples. Most of my video
133. lectures, by the way, will not be quite as long
as this one. We had a lot to cover today.
134. And there will be a short quiz after most
of my video lectures, a way for you to
135. rote test your understanding of the
material. Those will not be graded. At the
136. end of each unit, of which there are ten,
there will be a unit test. Those are
137. graded electronically, and that grade will
be the basis of your eligibility for a
138. certificate of completion of this course,
if you want one. Some of you may not want
139. a certificate. You may just wanna watch
the lectures, and that's absolutely fine.
140. This is all here on offer for you to use
in the way that best suits you.
141. A word about computers. Functional
computer literacy is a prerequisite for
142. this course. If you can't program, you're
not gonna be able to write the programs
143. that you will need to explore in the
homework. Now, I've designed the course so
144. that you can still pass it without doing
that and you can still get a flavor of the
145. concepts. But to get the full experience,
you really do need to be able to do the
146. homework. And there will be problems on
each exam that depend on your having done
147. the programming for the homework for that
unit. You're welcome to use any computer
148. programming language that you wish, modern
computer programming languages are all
149. Turing equivalents, so it shouldn't matter
what you use. What's gonna matter is what
150. comes out of your code, not the how well
commented it is or what style it has.
151. We're interested in what comes out and
that's what we'll be looking for in the
152. exams and the quizzes. Another related and
important point, there are thousands of
153. you, and among the thousands of you, there
are going to be dozens of favorite
154. programming languages, so there's no way
that we'll be able to help you debug your
155. code. You can post on the forum, and your