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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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    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 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
    that we'll be able to help you debug your
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    code. You can post on the forum, and your
    classmates will help you. Please do not
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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.