WEBVTT 99:59:59.999 --> 99:59:59.999 Hi this is Liz Bradley, I'm a Professor in the Computer Science department 99:59:59.999 --> 99:59:59.999 in the University of Colorado at Boulder and also on the external faculty of the 99:59:59.999 --> 99:59:59.999 Santa Fe Institute. My research interests are on nonlinear dynamics and chaos and 99:59:59.999 --> 99:59:59.999 in artificial intelligence, and I'm going to be your guide in this course on 99:59:59.999 --> 99:59:59.999 nonlinear dynamics and chaos. Here's an example of a nonlinear dynamical system. 99:59:59.999 --> 99:59:59.999 It's a double pendulum. Two pieces of aluminium and four ball bearings. Even 99:59:59.999 --> 99:59:59.999 though the system is very simple, it's behavior is very complicated. 99:59:59.999 --> 99:59:59.999 Moreover, this system is sensitively dependent on dynamical systems. If I 99:59:59.999 --> 99:59:59.999 started here, or here, the future evolution of the behavior will be very different. 99:59:59.999 --> 99:59:59.999 Even though the behavior of that device is very very complicated, there are some very 99:59:59.999 --> 99:59:59.999 strong patterns in that behavior, and the tandem of those patterns and the sensitivity 99:59:59.999 --> 99:59:59.999 is the hallmark of chaos. Now there's lots of words on this slide that we'll get 99:59:59.999 --> 99:59:59.999 into over the next ten weeks. I'll just give you some highlights here. 99:59:59.999 --> 99:59:59.999 A deterministic system is one that is not random. Cause and effect are linked and 99:59:59.999 --> 99:59:59.999 the current state determines the future state. 99:59:59.999 --> 99:59:59.999 A dynamic system (or a dynamical system), either are fine, is a system that evolves with time 99:59:59.999 --> 99:59:59.999 A nonlinear system is one where the relationships between the variables that 99:59:59.999 --> 99:59:59.999 matter are not linear. An example of a non linear system is the gas gauge in a car, 99:59:59.999 --> 99:59:59.999 at least in my car, where I fill up the tank, and then I drive a hundred miles and 99:59:59.999 --> 99:59:59.999 the needle barely moves. And then I drive another hundred miles and the needle. 99:59:59.999 --> 99:59:59.999 plummets. That's a nonlinear relationship between the level of gas in the tank 99:59:59.999 --> 99:59:59.999 and the position of the needle. Now non linear dynamics and chaos are not rare. 99:59:59.999 --> 99:59:59.999 Of all the systems in the universe that evolves with time, that's the outer 99:59:59.999 --> 99:59:59.999 ellipse in this Venn diagram, the vast majority of them are nonlinear. 99:59:59.999 --> 99:59:59.999 Indeed a famous mathematician refers to the study of nonlinear dynamics as the 99:59:59.999 --> 99:59:59.999 study of non-elephant animals. Now this is somewhat problematic, because the 99:59:59.999 --> 99:59:59.999 traditional training that we get in science, engineering and mathematics uses 99:59:59.999 --> 99:59:59.999 the assumption of linearity, and that's only a very small part of the picture. 99:59:59.999 --> 99:59:59.999 Now looking at the inner two ellipses on this Venn diagram conveys the point that 99:59:59.999 --> 99:59:59.999 the majority of nonlinear systems are chaotic, and so that's gonna play a big 99:59:59.999 --> 99:59:59.999 role in this course. And the equations that describe chaotic systems cannot be 99:59:59.999 --> 99:59:59.999 solved analytically, that is with a paper and pencil, rather we have to solve them 99:59:59.999 --> 99:59:59.999 with computers. And that is a large part of what distinguishes this course on 99:59:59.999 --> 99:59:59.999 nonlinear dynamics and chaos from most other courses on this topic area, 99:59:59.999 --> 99:59:59.999 including Steve Strogatz's great lectures which are on the web, and the courses on 99:59:59.999 --> 99:59:59.999 the complexity explorer website about this topic. We will focus not only on the 99:59:59.999 --> 99:59:59.999 mathematics, but also on the role of computation in the field. In this field, 99:59:59.999 --> 99:59:59.999 the computer is the lab instrument. This is experimental mathematics. And that's 99:59:59.999 --> 99:59:59.999 actually why the field of nonlinear dynamics only took off four decades ago 99:59:59.999 --> 99:59:59.999 Before that, there weren't computers to help us solve the equations. Now to 99:59:59.999 --> 99:59:59.999 succeed in this course, you'll need to understand the notion of a derivative, 99:59:59.999 --> 99:59:59.999 because dynamical systems are about change with time, and derivatives are the 99:59:59.999 --> 99:59:59.999 mathematics of change with time. You'll also need to be able to write simple 99:59:59.999 --> 99:59:59.999 computer programs. Basically, to translate simple mathematics formulas into code, run 99:59:59.999 --> 99:59:59.999 them, and plot the results, say on the axis of x versus t. There is no required 99:59:59.999 --> 99:59:59.999 computer language. You can use whichever programming language you want. 99:59:59.999 --> 99:59:59.999 And you're not gonna turn in your code in this course. We're interested in the 99:59:59.999 --> 99:59:59.999 results that come out of it. You'll also need to know about basic classical 99:59:59.999 --> 99:59:59.999 mechanics, the stuff that you get in first semester physics, like pendulums and 99:59:59.999 --> 99:59:59.999 masses on springs, and bodies pulling on each other, with GmM over r-squared kinds 99:59:59.999 --> 99:59:59.999 of forces. Speaking of GmM over r-squared, you may have seen this movie in the promo 99:59:59.999 --> 99:59:59.999 video that I made. This is movie taken by a camera on the Cassidy spacecraft as it 99:59:59.999 --> 99:59:59.999 flew by Saturn's moon, Hyperion. Hyperion is a very unusual shape and as a result of 99:59:59.999 --> 99:59:59.999 that shape, it tumbles chaotically. There's also chaos on how planets move 99:59:59.999 --> 99:59:59.999 through space, not just how they tumble. You may remember from Physics, that the 99:59:59.999 --> 99:59:59.999 solutions in those cases can only be conic sections, ellipses, parabolas and 99:59:59.999 --> 99:59:59.999 hyperbolas. As we will see, systems with three or more bodies can be chaotic. Now 99:59:59.999 --> 99:59:59.999 think about it, how many bodies are there in the solar system: lots more than two. 99:59:59.999 --> 99:59:59.999 Indeed several hundred years, the King of Sweden issued the challenge of a large 99:59:59.999 --> 99:59:59.999 cash prize to the person who could prove whether or not the solar system was stable 99:59:59.999 --> 99:59:59.999 in the long term, and that prize was never claimed. But the answer appeared in the 99:59:59.999 --> 99:59:59.999 1980s. Indeed the solar system is chaotic, although it is stable in a sense and we'll 99:59:59.999 --> 99:59:59.999 get back to that. So just some brief history of our field, it really dates back 99:59:59.999 --> 99:59:59.999 to Henri Poincare