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## ← Nonlinear 1.1 Introduction to nonlinear dynamics

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Showing Revision 12 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
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A nonlinear system is one where the
relationships between the variables that
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matter are not linear. An example of a non
linear system is the gas gauge in a car,
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at least in my car, where I fill up the
tank, and then I drive a hundred miles and
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the needle barely moves. And then I drive
another hundred miles and the needle.
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plummets. That's a nonlinear relationship
between the level of gas in the tank
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and the position of the needle. Now non
linear dynamics and chaos are not rare.
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Of all the systems in the universe that
evolves with time, that's the outer
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ellipse in this Venn diagram, the vast
majority of them are nonlinear.
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Indeed a famous mathematician refers to
the study of nonlinear dynamics as the
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study of non-elephant animals. Now this is
somewhat problematic, because the
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traditional training that we get in
science, engineering and mathematics uses
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the assumption of linearity, and that's
only a very small part of the picture.
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Now looking at the inner two ellipses on
this Venn diagram conveys the point that
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the majority of nonlinear systems are
chaotic, and so that's gonna play a big
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role in this course. And the equations
that describe chaotic systems cannot be
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solved analytically, that is with a paper
and pencil, rather we have to solve them
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with computers. And that is a large part
of what distinguishes this course on
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nonlinear dynamics and chaos from most
other courses on this topic area,
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including Steve Strogatz's great lectures
which are on the web, and the courses on
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topic. We will focus not only on the
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mathematics, but also on the role of
computation in the field. In this field,
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the computer is the lab instrument. This
is experimental mathematics. And that's
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actually why the field of nonlinear
dynamics only took off four decades ago
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Before that, there weren't computers to
help us solve the equations. Now to
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succeed in this course, you'll need to
understand the notion of a derivative,
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because dynamical systems are about change
with time, and derivatives are the
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mathematics of change with time. You'll
also need to be able to write simple
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computer programs. Basically, to translate
simple mathematics formulas into code, run
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them, and plot the results, say on the
axis of x versus t. There is no required
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computer language. You can use
whichever programming language you want.
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And you're not gonna turn in your code in
this course. We're interested in the
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results that come out of it. You'll also
need to know about basic classical
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mechanics, the stuff that you get in first
semester physics, like pendulums and
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masses on springs, and bodies pulling on
each other, with GmM over r-squared kinds
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of forces. Speaking of GmM over r-squared,
you may have seen this movie in the promo
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video that I made. This is movie taken by
a camera on the Cassidy spacecraft as it
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flew by Saturn's moon, Hyperion. Hyperion
is a very unusual shape and as a result of
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that shape, it tumbles chaotically.
There's also chaos on how planets move
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through space, not just how they tumble.
You may remember from Physics, that the
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solutions in those cases can only be conic
sections, ellipses, parabolas and
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hyperbolas. As we will see, systems with
three or more bodies can be chaotic. Now
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think about it, how many bodies are there
in the solar system: lots more than two.
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Indeed several hundred years, the King of
Sweden issued the challenge of a large
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cash prize to the person who could prove
whether or not the solar system was stable
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in the long term, and that prize was never
claimed. But the answer appeared in the
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1980s. Indeed the solar system is chaotic,
although it is stable in a sense and we'll
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get back to that. So just some brief
history of our field, it really dates back
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to Henri Poincare in the late 1800s. But
it really got going in the 1960s with Ed
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Lorentz's paper, called Deterministic Non
periodic Flow. Lorentz was the first
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person to recognize the patterns of chaos
and the sensitivity of the evolution of
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the system, within the context of those
patterns. In the 70s, this paper by Li and
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Yorke was the first to use the word
"chaos" in conjunction with this behavior.
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In the late 70s and 80s, the chaos cabal
at the University of California at Santa
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Cruz, got very interested in nonlinear
dynamics, and one of the problems that
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they approached it with was trying to beat
roulette, that is, modelling the path of a
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ball on a roulette wheel, and using that
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things really took off. And I should say,
of course, that I'm only cherry-picking a
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very small number of examples by lots of
smart people in a very active field.
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Nonlinear dynamics turns up all over the
place. Imagine an eddy in a creek, so a
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patch of swirling water on the surface of
a creek or a river, you can imagine
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dropping a wood chip in that patch of
water and watching its path from above,
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perhaps with a camera, and then dropping
another wood chip in that eddy at a
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slightly different point, and watching its
path. Those paths, they will trace out
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the patches of swirling water in that eddy
in different order, but if you did a time
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lapse photograph of their paths, they
would both trace out the same eddy.
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Weather is nonlinear and chaotic. You may
have heard of the butterfly effect.
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A butterfly flapping its wings setting off
a hurricane a week later, a thousand miles
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away. Again, small change, large effect,
sensitive dependence on initial conditions
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Marine invertebrates actually make use of
chaotic mixing in the water around them
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during spawning, and I'm interested in
exploiting chaotic mixing to design better
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fuel injectors in cars. Nonlinear and
chaotic dynamics also turns up in driven
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nonlinear oscillators, like the pendulum
that I showed you, like the human heart
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which is normally kind of mostly periodic
but, can go into a chaotic state called
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ventricular fibrillation and as you saw
with the example of Hyperion, there's a
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lot of nonlinear and chaotic dynamics in
classical mechanics ranging from the three
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body problem and how black holes move
around each other. And nonlinear and
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chaotic dynamics turns up in lots and lots
of other fields, including, certainly,
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things that you are interested in. So as I
hope you can see, nonlinear and chaotic
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dynamics are not an academic oddity. They
are widespread, and they are fascinating,
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and I hope that you will get infected by
some of that fascination over the course
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of the next ten weeks. There are other
fascinating courses on the Complexity
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Explorer website including Dave Feldman's
course on the same topic area that only
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assumes knowledge of high school algebra,
and Melanie Mitchell's wonderful course on
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complexity. The difference between
complexity and chaos actually bears a
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little bit of explanation. Put perhaps too
simply, you can think of chaos as
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complicated behavior from simple systems,
like my pendulum. And you can think of
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that are very complicated but have simple
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behavior. Again, that is too pat but the
idea is generally right. So, a thousand
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fish forming a single school. Now, some
logistics. There are several thousand of
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you and one of me. We have an email
address for this course but it can very
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rapidly get overwhelmed. Please do not use
my own personal email address, or that of
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the TA, for course-related communications.
That thousands-to-one ratios is one of the
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major issues with MOOCs like this one.
Part of the way we plan to work around
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that is with an electronic forum. This is
not just to take a load off the course
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staff, it's also to solve one of the other
problems with MOOCs, which is, instead of
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being in a traditional classroom, everyone
taking this course is working by themselves
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all over the world in all sorts of time
zones. And we hope to use the forum to
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help with that. So if you've a question,
look on the forum. Someone else may have
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posted that question already. If not,
post it yourself. If someone has posted an
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question that you know the answer to, or
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also use the forum, by the way, to post
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announcements, like there's a bug in the
problem set, or I've just posted a whole
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new unit, or, the New York Times has an
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I'll also post discussion questions and
answers for topics that may interest some
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people in the course, if somebody wants to
go deeper into something or sideways along
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a tangent, that's where the forum can
play a role.
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Here's another piece of technology that
can help.
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There's no textbooks for this course. I'm
pulling together material from many many
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different sources, including a substantial
amount from my own work, papers that I've
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read, talks that I've heard at conferences
and so on and so forth.
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These video lectures are short, self-
contained summaries of each topic. I use
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the Supplementary Materials page to
supplement those summaries. So if you want
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to dig more deeply into something I
mentioned, or you'd like some background
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material, or, you wanna read the original
paper that I mentioned. This is where you
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should look. In the next segment of this
course, we'll start digging into some
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ideas and mathematics and plots and
computer examples. Most of my video
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lectures, by the way, are not quite as long
as this one. We had a lot to cover today.
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And there will be a short quiz after most
of my video lectures, a way for you to
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rote test your understanding of the
material. Those will not be graded. At the
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end of each unit, of which there are ten,
there will be a unit test. Those are
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be the basis of your eligibility for a
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certificate of completion for this course,
if you want one. Some of you may not want
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a certificate. You may just wanna watch
the lectures, and that's absolutely fine.
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This is all here on offer for you to use
in the way that best suits you.
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computer literacy is a prerequisite for
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this course. If you can't program, you're
not gonna be able to write the programs
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that you will need to explore in the
homework. Now, I've designed the course so
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that you can still pass it without doing
that and you can still get a flavor of the
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concepts. But to get the full experience,
you really do need to be able to do the
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homework. And there will be problems on
each exam that depend on your having done
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the programming for the homework for that
unit. You're welcome to use any computer
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programming language that you wish, modern
computer programming languages are all
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Turing equivalents, so it shouldn't matter
what you use. What's gonna matter is what
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comes out of your code, not the how well
commented it is or what style it has.
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We're interested in what comes out and
that's what we'll be looking for in the
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exams and the quizzes. Another related and
important point, there are thousands of
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you, and among the thousands of you, there
are going to be dozens of favorite
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programming languages, so there's no way
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code. You can post on the forum, and your
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just post entire solutions on the forum
and ask, "Where's the bug?" We have
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chosen Matlab as the program in which we
will post our solutions, because it's
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pretty widespread and pretty simple. It's
a good lingua franca for that purpose. If
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you've never encountered Matlab, you may
want to look over one of the many
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tutorials that are available on the web
for the basic syntax for that language so
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that you can understand our solutions.