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Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Hi this is Liz Bradley, I'm a Professor\Nin the Computer Science department
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,in the University of Colorado at Boulder\Nand also on the external faculty of the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Santa Fe Institute. My research interests\Nare on nonlinear dynamics and chaos and
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,in artificial intelligence, and I'm going\Nto be your guide in this course on
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,nonlinear dynamics and chaos. Here's an\Nexample of a nonlinear dynamical system.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,It's a double pendulum. Two pieces of\Naluminium and four ball bearings. Even
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,though the system is very simple, it's\Nbehavior is very complicated.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Moreover, this system is sensitively\Ndependent on dynamical systems. If I
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,started here, or here, the future evolution\Nof the behavior will be very different.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Even though the behavior of that device is\Nvery very complicated, there are some very
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,strong patterns in that behavior, and the\Ntandem of those patterns and the sensitivity
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,is the hallmark of chaos. Now there's \Nlots of words on this slide that we'll get
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,into over the next ten weeks. I'll just\Ngive you some highlights here.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,A deterministic system is one that is not\Nrandom. Cause and effect are linked and
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the current state determines the future\Nstate.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,A dynamic system (or a dynamical system),\Neither are fine, is a system that evolves with time
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,A nonlinear system is one where the\Nrelationships between the variables that
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,matter are not linear. An example of a non\Nlinear system is the gas gauge in a car,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,at least in my car, where I fill up the\Ntank, and then I drive a hundred miles and
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the needle barely moves. And then I drive \Nanother hundred miles and the needle.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,plummets. That's a nonlinear relationship \Nbetween the level of gas in the tank
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and the position of the needle. Now non\Nlinear dynamics and chaos are not rare.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Of all the systems in the universe that\Nevolves with time, that's the outer
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,ellipse in this Venn diagram, the vast\Nmajority of them are nonlinear.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Indeed a famous mathematician refers to\Nthe study of nonlinear dynamics as the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,study of non-elephant animals. Now this is\Nsomewhat problematic, because the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,traditional training that we get in\Nscience, engineering and mathematics uses
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the assumption of linearity, and that's\Nonly a very small part of the picture.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Now looking at the inner two ellipses on\Nthis Venn diagram conveys the point that
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the majority of nonlinear systems are\Nchaotic, and so that's gonna play a big
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,role in this course. And the equations\Nthat describe chaotic systems cannot be
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,solved analytically, that is with a paper\Nand pencil, rather we have to solve them
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,with computers. And that is a large part\Nof what distinguishes this course on
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,nonlinear dynamics and chaos from most\Nother courses on this topic area,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,including Steve Strogatz's great lectures\Nwhich are on the web, and the courses on
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the complexity explorer website about this\Ntopic. We will focus not only on the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,mathematics, but also on the role of\Ncomputation in the field. In this field,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the computer is the lab instrument. This\Nis experimental mathematics. And that's
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,actually why the field of nonlinear\Ndynamics only took off four decades ago
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Before that, there weren't computers to \Nhelp us solve the equations. Now to
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,succeed in this course, you'll need to\Nunderstand the notion of a derivative,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,because dynamical systems are about change\Nwith time, and derivatives are the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,mathematics of change with time. You'll\Nalso need to be able to write simple
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,computer programs. Basically, to translate\Nsimple mathematics formulas into code, run
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,them, and plot the results, say on the\Naxis of x versus t. There is no required
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,computer language. You can use\Nwhichever programming language you want.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,And you're not gonna turn in your code in\Nthis course. We're interested in the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,results that come out of it. You'll also\Nneed to know about basic classical
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,mechanics, the stuff that you get in first\Nsemester physics, like pendulums and
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,masses on springs, and bodies pulling on\Neach other, with GmM over r-squared kinds
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,of forces. Speaking of GmM over r-squared,\Nyou may have seen this movie in the promo
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,video that I made. This is movie taken by \Na camera on the Cassidy spacecraft as it
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,flew by Saturn's moon, Hyperion. Hyperion\Nis a very unusual shape and as a result of
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,that shape, it tumbles chaotically.\NThere's also chaos on how planets move
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,through space, not just how they tumble.\NYou may remember from Physics, that the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,solutions in those cases can only be conic\Nsections, ellipses, parabolas and
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,hyperbolas. As we will see, systems with\Nthree or more bodies can be chaotic. Now
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,think about it, how many bodies are there\Nin the solar system: lots more than two.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Indeed several hundred years, the King of\NSweden issued the challenge of a large
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,cash prize to the person who could prove\Nwhether or not the solar system was stable
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,in the long term, and that prize was never\Nclaimed. But the answer appeared in the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,1980s. Indeed the solar system is chaotic,\Nalthough it is stable in a sense and we'll
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,get back to that. So just some brief\Nhistory of our field, it really dates back
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,to Henri Poincare