Return to Video

Nonlinear 1.1 Introduction to nonlinear dynamics

  • 0:03 - 0:07
    Hi this is Liz Bradley, I'm a Professor
    in the Computer Science department
  • 0:08 - 0:11
    at the University of Colorado at Boulder
    and also on the external faculty of the
  • 0:12 - 0:17
    Santa Fe Institute. My research interests
    are in nonlinear dynamics and chaos and
  • 0:17 - 0:21
    in artificial intelligence, and I'm going
    to be your guide during this course on
  • 0:21 - 0:25
    nonlinear dynamics and chaos. Here's an
    example of a nonlinear dynamical system.
  • 0:27 - 0:31
    It's a double pendulum. Two pieces of
    aluminium and four ball bearings. Even
  • 0:31 - 0:36
    though the system is physically very
    simple, it's behavior is very complicated.
  • 0:42 - 0:46
    Moreover, this system is sensitively
    dependent on initial conditions. If I
  • 0:46 - 0:51
    started here, or here, the future evolution
    of the behavior will be very different.
  • 1:00 - 1:04
    Even though the behavior of that device is
    very very complicated, there are some very
  • 1:04 - 1:08
    strong patterns in that behavior, and the
    tandem of those patterns and the sensitivity
  • 1:08 - 1:14
    is the hallmark of chaos. Now there's
    lots of words on this slide that we'll get
  • 1:14 - 1:17
    into over the next ten weeks. I'll just
    give you some highlights here.
  • 1:17 - 1:21
    A deterministic system is one that is not
    random. Cause and effect are linked and
  • 1:21 - 1:24
    the current state determines the future
  • 1:24 - 1:28
    A dynamic system (or a dynamical system),
    either are fine, is a system that evolves with time
  • Not Synced
    A nonlinear system is one where the
    relationships between the variables that
  • Not Synced
    matter are not linear. An example of a non
    linear system is the gas gauge in a car,
  • Not Synced
    at least in my car, where I fill up the
    tank, and then I drive a hundred miles and
  • Not Synced
    the needle barely moves. And then I drive
    another hundred miles and the needle.
  • Not Synced
    plummets. That's a nonlinear relationship
    between the level of gas in the tank
  • Not Synced
    and the position of the needle. Now non
    linear dynamics and chaos are not rare.
  • Not Synced
    Of all the systems in the universe that
    evolves with time, that's the outer
  • Not Synced
    ellipse in this Venn diagram, the vast
    majority of them are nonlinear.
  • Not Synced
    Indeed a famous mathematician refers to
    the study of nonlinear dynamics as the
  • Not Synced
    study of non-elephant animals. Now this is
    somewhat problematic, because the
  • Not Synced
    traditional training that we get in
    science, engineering and mathematics uses
  • Not Synced
    the assumption of linearity, and that's
    only a very small part of the picture.
  • Not Synced
    Now looking at the inner two ellipses on
    this Venn diagram conveys the point that
  • Not Synced
    the majority of nonlinear systems are
    chaotic, and so that's gonna play a big
  • Not Synced
    role in this course. And the equations
    that describe chaotic systems cannot be
  • Not Synced
    solved analytically, that is with a paper
    and pencil, rather we have to solve them
  • Not Synced
    with computers. And that is a large part
    of what distinguishes this course on
  • Not Synced
    nonlinear dynamics and chaos from most
    other courses on this topic area,
  • Not Synced
    including Steve Strogatz's great lectures
    which are on the web, and the courses on
  • Not Synced
    the complexity explorer website about this
    topic. We will focus not only on the
  • Not Synced
    mathematics, but also on the role of
    computation in the field. In this field,
  • Not Synced
    the computer is the lab instrument. This
    is experimental mathematics. And that's
  • Not Synced
    actually why the field of nonlinear
    dynamics only took off four decades ago
  • Not Synced
    Before that, there weren't computers to
    help us solve the equations. Now to
  • Not Synced
    succeed in this course, you'll need to
    understand the notion of a derivative,
  • Not Synced
    because dynamical systems are about change
    with time, and derivatives are the
  • Not Synced
    mathematics of change with time. You'll
    also need to be able to write simple
  • Not Synced
    computer programs. Basically, to translate
    simple mathematics formulas into code, run
  • Not Synced
    them, and plot the results, say on the
    axis of x versus t. There is no required
  • Not Synced
    computer language. You can use
    whichever programming language you want.
  • Not Synced
    And you're not gonna turn in your code in
    this course. We're interested in the
  • Not Synced
    results that come out of it. You'll also
    need to know about basic classical
  • Not Synced
    mechanics, the stuff that you get in first
    semester physics, like pendulums and
  • Not Synced
    masses on springs, and bodies pulling on
    each other, with GmM over r-squared kinds
  • Not Synced
    of forces. Speaking of GmM over r-squared,
    you may have seen this movie in the promo
  • Not Synced
    video that I made. This is movie taken by
    a camera on the Cassidy spacecraft as it
  • Not Synced
    flew by Saturn's moon, Hyperion. Hyperion
    is a very unusual shape and as a result of
  • Not Synced
    that shape, it tumbles chaotically.
    There's also chaos on how planets move
  • Not Synced
    through space, not just how they tumble.
    You may remember from Physics, that the
  • Not Synced
    solutions in those cases can only be conic
    sections, ellipses, parabolas and
  • Not Synced
    hyperbolas. As we will see, systems with
    three or more bodies can be chaotic. Now
  • Not Synced
    think about it, how many bodies are there
    in the solar system: lots more than two.
  • Not Synced
    Indeed several hundred years, the King of
    Sweden issued the challenge of a large
  • Not Synced
    cash prize to the person who could prove
    whether or not the solar system was stable
  • Not Synced
    in the long term, and that prize was never
    claimed. But the answer appeared in the
  • Not Synced
    1980s. Indeed the solar system is chaotic,
    although it is stable in a sense and we'll
  • Not Synced
    get back to that. So just some brief
    history of our field, it really dates back
  • Not Synced
    to Henri Poincare in the late 1800s. But
    it really got going in the 1960s with Ed
  • Not Synced
    Lorentz's paper, called Deterministic Non
    periodic Flow. Lorentz was the first
  • Not Synced
    person to recognize the patterns of chaos
    and the sensitivity of the evolution of
  • Not Synced
    the system, within the context of those
    patterns. In the 70s, this paper by Li and
  • Not Synced
    Yorke was the first to use the word
    "chaos" in conjunction with this behavior.
  • Not Synced
    In the late 70s and 80s, the chaos cabal
    at the University of California at Santa
  • Not Synced
    Cruz, got very interested in nonlinear
    dynamics, and one of the problems that
  • Not Synced
    they approached it with was trying to beat
    roulette, that is, modelling the path of a
  • Not Synced
    ball on a roulette wheel, and using that
    information to advantage. After this,
  • Not Synced
    things really took off. And I should say,
    of course, that I'm only cherry-picking a
  • Not Synced
    very small number of examples by lots of
    smart people in a very active field.
  • Not Synced
    Nonlinear dynamics turns up all over the
    place. Imagine an eddy in a creek, so a
  • Not Synced
    patch of swirling water on the surface of
    a creek or a river, you can imagine
  • Not Synced
    dropping a wood chip in that patch of
    water and watching its path from above,
  • Not Synced
    perhaps with a camera, and then dropping
    another wood chip in that eddy at a
  • Not Synced
    slightly different point, and watching its
    path. Those paths, they will trace out
  • Not Synced
    the patches of swirling water in that eddy
    in different order, but if you did a time
  • Not Synced
    lapse photograph of their paths, they
    would both trace out the same eddy.
  • Not Synced
    Weather is nonlinear and chaotic. You may
    have heard of the butterfly effect.
  • Not Synced
    A butterfly flapping its wings setting off
    a hurricane a week later, a thousand miles
  • Not Synced
    away. Again, small change, large effect,
    sensitive dependence on initial conditions
  • Not Synced
    Marine invertebrates actually make use of
    chaotic mixing in the water around them
  • Not Synced
    during spawning, and I'm interested in
    exploiting chaotic mixing to design better
  • Not Synced
    fuel injectors in cars. Nonlinear and
    chaotic dynamics also turns up in driven
  • Not Synced
    nonlinear oscillators, like the pendulum
    that I showed you, like the human heart
  • Not Synced
    which is normally kind of mostly periodic
    but, can go into a chaotic state called
  • Not Synced
    ventricular fibrillation and as you saw
    with the example of Hyperion, there's a
  • Not Synced
    lot of nonlinear and chaotic dynamics in
    classical mechanics ranging from the three
  • Not Synced
    body problem and how black holes move
    around each other. And nonlinear and
  • Not Synced
    chaotic dynamics turns up in lots and lots
    of other fields, including, certainly,
  • Not Synced
    things that you are interested in. So as I
    hope you can see, nonlinear and chaotic
  • Not Synced
    dynamics are not an academic oddity. They
    are widespread, and they are fascinating,
  • Not Synced
    and I hope that you will get infected by
    some of that fascination over the course
  • Not Synced
    of the next ten weeks. There are other
    fascinating courses on the Complexity
  • Not Synced
    Explorer website including Dave Feldman's
    course on the same topic area that only
  • Not Synced
    assumes knowledge of high school algebra,
    and Melanie Mitchell's wonderful course on
  • Not Synced
    complexity. The difference between
    complexity and chaos actually bears a
  • Not Synced
    little bit of explanation. Put perhaps too
    simply, you can think of chaos as
  • Not Synced
    complicated behavior from simple systems,
    like my pendulum. And you can think of
  • Not Synced
    complexity science as addressing systems
    that are very complicated but have simple
  • Not Synced
    behavior. Again, that is too pat but the
    idea is generally right. So, a thousand
  • Not Synced
    fish forming a single school. Now, some
    logistics. There are several thousand of
  • Not Synced
    you and one of me. We have an email
    address for this course but it can very
  • Not Synced
    rapidly get overwhelmed. Please do not use
    my own personal email address, or that of
  • Not Synced
    the TA, for course-related communications.
    That thousands-to-one ratios is one of the
  • Not Synced
    major issues with MOOCs like this one.
    Part of the way we plan to work around
  • Not Synced
    that is with an electronic forum. This is
    not just to take a load off the course
  • Not Synced
    staff, it's also to solve one of the other
    problems with MOOCs, which is, instead of
  • Not Synced
    being in a traditional classroom, everyone
    taking this course is working by themselves
  • Not Synced
    all over the world in all sorts of time
    zones. And we hope to use the forum to
  • Not Synced
    help with that. So if you've a question,
    look on the forum. Someone else may have
  • Not Synced
    posted that question already. If not,
    post it yourself. If someone has posted an
  • Not Synced
    answer, look at that answer. If you see a
    question that you know the answer to, or
  • Not Synced
    you think you do, offer your answer. I'll
    also use the forum, by the way, to post
  • Not Synced
    announcements, like there's a bug in the
    problem set, or I've just posted a whole
  • Not Synced
    new unit, or, the New York Times has an
    article about the stuff I just talked about.
  • Not Synced
    I'll also post discussion questions and
    answers for topics that may interest some
  • Not Synced
    people in the course, if somebody wants to
    go deeper into something or sideways along
  • Not Synced
    a tangent, that's where the forum can
    play a role.
  • Not Synced
    Here's another piece of technology that
    can help.
  • Not Synced
    There's no textbooks for this course. I'm
    pulling together material from many many
  • Not Synced
    different sources, including a substantial
    amount from my own work, papers that I've
  • Not Synced
    read, talks that I've heard at conferences
    and so on and so forth.
  • Not Synced
    These video lectures are short, self-
    contained summaries of each topic. I use
  • Not Synced
    the Supplementary Materials page to
    supplement those summaries. So if you want
  • Not Synced
    to dig more deeply into something I
    mentioned, or you'd like some background
  • Not Synced
    material, or, you wanna read the original
    paper that I mentioned. This is where you
  • Not Synced
    should look. In the next segment of this
    course, we'll start digging into some
  • Not Synced
    ideas and mathematics and plots and
    computer examples. Most of my video
  • Not Synced
    lectures, by the way, are not quite as long
    as this one. We had a lot to cover today.
  • Not Synced
    And there will be a short quiz after most
    of my video lectures, a way for you to
  • Not Synced
    rote test your understanding of the
    material. Those will not be graded. At the
  • Not Synced
    end of each unit, of which there are ten,
    there will be a unit test. Those are
  • Not Synced
    graded electronically, and that grade will
    be the basis of your eligibility for a
  • Not Synced
    certificate of completion for this course,
    if you want one. Some of you may not want
  • Not Synced
    a certificate. You may just wanna watch
    the lectures, and that's absolutely fine.
  • Not Synced
    This is all here on offer for you to use
    in the way that best suits you.
  • Not Synced
    A word about computers. Functional
    computer literacy is a prerequisite for
  • Not Synced
    this course. If you can't program, you're
    not gonna be able to write the programs
  • Not Synced
    that you will need to explore in the
    homework. Now, I've designed the course so
  • Not Synced
    that you can still pass it without doing
    that and you can still get a flavor of the
  • Not Synced
    concepts. But to get the full experience,
    you really do need to be able to do the
  • Not Synced
    homework. And there will be problems on
    each exam that depend on your having done
  • Not Synced
    the programming for the homework for that
    unit. You're welcome to use any computer
  • Not Synced
    programming language that you wish, modern
    computer programming languages are all
  • Not Synced
    Turing equivalents, so it shouldn't matter
    what you use. What's gonna matter is what
  • Not Synced
    comes out of your code, not the how well
    commented it is or what style it has.
  • Not Synced
    We're interested in what comes out and
    that's what we'll be looking for in the
  • Not Synced
    exams and the quizzes. Another related and
    important point, there are thousands of
  • Not Synced
    you, and among the thousands of you, there
    are going to be dozens of favorite
  • Not Synced
    programming languages, so there's no way
    that we'll be able to help you debug your
  • Not Synced
    code. You can post on the forum, and your
    classmates will help you. Please do not
  • Not Synced
    just post entire solutions on the forum
    and ask, "Where's the bug?" We have
  • Not Synced
    chosen Matlab as the program in which we
    will post our solutions, because it's
  • Not Synced
    pretty widespread and pretty simple. It's
    a good lingua franca for that purpose. If
  • Not Synced
    you've never encountered Matlab, you may
    want to look over one of the many
  • Not Synced
    tutorials that are available on the web
    for the basic syntax for that language so
  • Not Synced
    that you can understand our solutions.
Nonlinear 1.1 Introduction to nonlinear dynamics
Video Language:
Complexity Explorer
Nonlinear Dynamics

English subtitles

Revisions Compare revisions