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← Nonlinear 1.1 Introduction to nonlinear dynamics

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Showing Revision 11 created 07/20/2016 by Steven Gunawan.

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    Hi this is Liz Bradley, I'm a Professor
    in the Computer Science department
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    at the University of Colorado at Boulder
    and also on the external faculty of the
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    Santa Fe Institute. My research interests
    are on nonlinear dynamics and chaos and
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    in artificial intelligence, and I'm going
    to be your guide in this course on
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    nonlinear dynamics and chaos. Here's an
    example of a nonlinear dynamical system.
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    It's a double pendulum. Two pieces of
    aluminium and four ball bearings. Even
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    though the system is very simple, it's
    behavior is very complicated.
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    Moreover, this system is sensitively
    dependent on dynamical systems. If I
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    started here, or here, the future evolution
    of the behavior will be very different.
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    Even though the behavior of that device is
    very very complicated, there are some very
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    strong patterns in that behavior, and the
    tandem of those patterns and the sensitivity
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    is the hallmark of chaos. Now there's
    lots of words on this slide that we'll get
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    into over the next ten weeks. I'll just
    give you some highlights here.
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    A deterministic system is one that is not
    random. Cause and effect are linked and
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    the current state determines the future
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    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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    the complexity explorer website about this
    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
    information to advantage. After this,
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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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    complexity science as addressing systems
    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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    answer, look at that answer. If you see a
    question that you know the answer to, or
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    you think you do, offer your answer. I'll
    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
    article about the stuff I just talked about.
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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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    graded electronically, and that grade will
    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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    A word about computers. Functional
    computer literacy is a