9:59:59.000,9:59:59.000 Hi this is Liz Bradley, I'm a Professor[br]in the Computer Science department 9:59:59.000,9:59:59.000 in the University of Colorado at Boulder[br]and also on the external faculty of the 9:59:59.000,9:59:59.000 Santa Fe Institute. My research interests[br]are on nonlinear dynamics and chaos and 9:59:59.000,9:59:59.000 in artificial intelligence, and I'm going[br]to be your guide in this course on 9:59:59.000,9:59:59.000 nonlinear dynamics and chaos. Here's an[br]example of a nonlinear dynamical system. 9:59:59.000,9:59:59.000 It's a double pendulum. Two pieces of[br]aluminium and four ball bearings. Even 9:59:59.000,9:59:59.000 though the system is very simple, it's[br]behavior is very complicated. 9:59:59.000,9:59:59.000 Moreover, this system is sensitively[br]dependent on dynamical systems. If I 9:59:59.000,9:59:59.000 started here, or here, the future evolution[br]of the behavior will be very different. 9:59:59.000,9:59:59.000 Even though the behavior of that device is[br]very very complicated, there are some very 9:59:59.000,9:59:59.000 strong patterns in that behavior, and the[br]tandem of those patterns and the sensitivity 9:59:59.000,9:59:59.000 is the hallmark of chaos. Now there's [br]lots of words on this slide that we'll get 9:59:59.000,9:59:59.000 into over the next ten weeks. I'll just[br]give you some highlights here. 9:59:59.000,9:59:59.000 A deterministic system is one that is not[br]random. Cause and effect are linked and 9:59:59.000,9:59:59.000 the current state determines the future[br]state. 9:59:59.000,9:59:59.000 A dynamic system (or a dynamical system),[br]either are fine, is a system that evolves with time 9:59:59.000,9:59:59.000 A nonlinear system is one where the[br]relationships between the variables that 9:59:59.000,9:59:59.000 matter are not linear. An example of a non[br]linear system is the gas gauge in a car, 9:59:59.000,9:59:59.000 at least in my car, where I fill up the[br]tank, and then I drive a hundred miles and 9:59:59.000,9:59:59.000 the needle barely moves. And then I drive [br]another hundred miles and the needle. 9:59:59.000,9:59:59.000 plummets. That's a nonlinear relationship [br]between the level of gas in the tank 9:59:59.000,9:59:59.000 and the position of the needle. Now non[br]linear dynamics and chaos are not rare. 9:59:59.000,9:59:59.000 Of all the systems in the universe that[br]evolves with time, that's the outer 9:59:59.000,9:59:59.000 ellipse in this Venn diagram, the vast[br]majority of them are nonlinear. 9:59:59.000,9:59:59.000 Indeed a famous mathematician refers to[br]the study of nonlinear dynamics as the 9:59:59.000,9:59:59.000 study of non-elephant animals. Now this is[br]somewhat problematic, because the 9:59:59.000,9:59:59.000 traditional training that we get in[br]science, engineering and mathematics uses 9:59:59.000,9:59:59.000 the assumption of linearity, and that's[br]only a very small part of the picture. 9:59:59.000,9:59:59.000 Now looking at the inner two ellipses on[br]this Venn diagram conveys the point that 9:59:59.000,9:59:59.000 the majority of nonlinear systems are[br]chaotic, and so that's gonna play a big 9:59:59.000,9:59:59.000 role in this course. And the equations[br]that describe chaotic systems cannot be 9:59:59.000,9:59:59.000 solved analytically, that is with a paper[br]and pencil, rather we have to solve them 9:59:59.000,9:59:59.000 with computers. And that is a large part[br]of what distinguishes this course on 9:59:59.000,9:59:59.000 nonlinear dynamics and chaos from most[br]other courses on this topic area, 9:59:59.000,9:59:59.000 including Steve Strogatz's great lectures[br]which are on the web, and the courses on 9:59:59.000,9:59:59.000 the complexity explorer website about this[br]topic. We will focus not only on the 9:59:59.000,9:59:59.000 mathematics, but also on the role of[br]computation in the field. In this field, 9:59:59.000,9:59:59.000 the computer is the lab instrument. This[br]is experimental mathematics. And that's 9:59:59.000,9:59:59.000 actually why the field of nonlinear[br]dynamics only took off four decades ago 9:59:59.000,9:59:59.000 Before that, there weren't computers to [br]help us solve the equations. Now to 9:59:59.000,9:59:59.000 succeed in this course, you'll need to[br]understand the notion of a derivative, 9:59:59.000,9:59:59.000 because dynamical systems are about change[br]with time, and derivatives are the 9:59:59.000,9:59:59.000 mathematics of change with time. You'll[br]also need to be able to write simple 9:59:59.000,9:59:59.000 computer programs. Basically, to translate[br]simple mathematics formulas into code, run 9:59:59.000,9:59:59.000 them, and plot the results, say on the[br]axis of x versus t. There is no required 9:59:59.000,9:59:59.000 computer language. You can use[br]whichever programming language you want. 9:59:59.000,9:59:59.000 And you're not gonna turn in your code in[br]this course. We're interested in the 9:59:59.000,9:59:59.000 results that come out of it. You'll also[br]need to know about basic classical 9:59:59.000,9:59:59.000 mechanics, the stuff that you get in first[br]semester physics, like pendulums and 9:59:59.000,9:59:59.000 masses on springs, and bodies pulling on[br]each other, with GmM over r-squared kinds 9:59:59.000,9:59:59.000 of forces. Speaking of GmM over r-squared,[br]you may have seen this movie in the promo 9:59:59.000,9:59:59.000 video that I made. This is movie taken by [br]a camera on the Cassidy spacecraft as it 9:59:59.000,9:59:59.000 flew by Saturn's moon, Hyperion. Hyperion[br]is a very unusual shape and as a result of 9:59:59.000,9:59:59.000 that shape, it tumbles chaotically.[br]There's also chaos on how planets move 9:59:59.000,9:59:59.000 through space, not just how they tumble.[br]You may remember from Physics, that the 9:59:59.000,9:59:59.000 solutions in those cases can only be conic[br]sections, ellipses, parabolas and 9:59:59.000,9:59:59.000 hyperbolas. As we will see, systems with[br]three or more bodies can be chaotic. Now 9:59:59.000,9:59:59.000 think about it, how many bodies are there[br]in the solar system: lots more than two. 9:59:59.000,9:59:59.000 Indeed several hundred years, the King of[br]Sweden issued the challenge of a large 9:59:59.000,9:59:59.000 cash prize to the person who could prove[br]whether or not the solar system was stable 9:59:59.000,9:59:59.000 in the long term, and that prize was never[br]claimed. But the answer appeared in the 9:59:59.000,9:59:59.000 1980s. Indeed the solar system is chaotic,[br]although it is stable in a sense and we'll 9:59:59.000,9:59:59.000 get back to that. So just some brief[br]history of our field, it really dates back 9:59:59.000,9:59:59.000 to Henri Poincare