[MUSIC] Now, let's go back and take a look at it. First of all, we have the so called term that I'm sure most of you have heard sensitivity with respect to initial conditions. That is captured by that incredibly small region in A, which what, can expand and expand and expand, and you can have wild, wildly different futures, even though they started very, very close to each other. Not right away, but eventually they're going to start separating from each other. [COUGH] So, that is the source, the cause of chaotic dynamics, and to translate this into terms that you can use to take a look what's happening on the stock market. Look at what's happening in physics, what's happening any other place. Let's take a look at what are the principles. The principles are the following. The first one is expansion. Notice what happened. Define where in the a region I'm going to hit the c region. I had a very small region. We computed it. We computed where I have to be in the a region so I hit one boundary of c. Where I have to be so it hits the other region. And I ended up with that very very small little region, that first one, and what did the dynamic do, it expanded it. So, what's necessary, in chaotic dynamics is expansion. Contraction usually gives you stability. Stability is predictability. Expansion is crucial or lack of predictability for this random appearing type behavior, but that's not enough. If I have expansion, and I just keep moving in one direction and expanding, it's predictable, nothing's interesting. Where something is interesting is when you come back, is when you comeback that's what makes it interesting. So if I come back, I'm starting at an initial point and if I come back very very close to it. What do we know what the expansion is going to do. It's gonna make it do something very very different. So this mix like rolling a dice or something like that. This mix, that makes it look random, is a combination between expansion and recurrurence. I'm gonna give you a couple of examples to show you where this crops up. But expansion and recurrence, expansion and recurrurence does not, it does not state that you have chaotic behavior. It's the other way around. When you have chaotic behavior expect that there is expansion and recurrurence hiding inside the situation. So if you see expansion and recurrurence you should at least look, examine, try to see whether or not there is chaotic effects. Let's start with what you do early in the morning. That early morning coffee, I don't know about you but for me it's a necessity of life. There is a statement that a mathematician is a machine, that converts smart coffee into equations and I definitely am a coffee drinker. So what you do in the morning, you sit down there you put maybe your sugar, or you put maybe a little cream inside there and what do you do is you take your spoon, right? And that spoon, look at that spoon, you stir it. You have a deterministic motion, but you're stirring, and that stirring, take a look at the spoon, it's spreading everything out. You have an expansion effect, and the cup does what? The cup forces you to have a recurrurence. And so one of the places where we see chaoticita effects, is early morning coffee when you're stirring the coffee, or you're stirring your paint when you wanna redo your wall. Any of these types situations. Here's another one. It's a project that I worked on. I started off by talking about satellites up in outer space and part of my work is on mathematical astronomy. In mathematical astronomy we have a object up there called the ISS, the International Space Station. We also have in about the same orbit, 3 million pieces of junk, garbage. Ocean blows up, someone loses their wrench, maybe toss out a beer can. Oh, no, they don't do that, but there's junk up there floating around, and a piece of junk the size of my fist, if it hit the ISS, goodbye. That means they have to understand precisely where each of the three million pieces of junk are to try to save the space station. Indeed, every so often, these Astronauts have to get into a escape type mode, just in case one of these pieces of junk hits the ISS. Wait a minute, what's wrong with that? Newton has these equations. He has these equations of motion which are deterministic so what's so difficult? Chaotic dynamics, where yes, it's deterministic, but it can be random in appearance. Let's see why. First of all, the Earth is not a perfect sphere. We´ve got mountains, we have oceans, it has mid age spread, we´re wider at the Equator then we are at the top. And so, where we´re wider and have more mass, of the Earth. That´s a stronger pull, on the satellites, on the piece of junk that´s coming around, and what does that do? That gives you the expansion effect. And recurrence, hey, you know where the recurrurence is [LAUGH], it's going around the Earth. So a satellite now, much like that morning coffee, a molecule rather than a coffee cup is subjected to expansion and recurrurence. And so what we're going to have is a chaotic type behavior. What we would like is to have about 72 hours advance notice to move the ISS, and the reason for this is clear. There's not a local gas station there to load up once you run out of fuel, and so therefore, what we wanna do is we wanna conserve it. So if we have 72 hours, we can move that space station very slowly conserving all the fuel etc. But recurrent methods because of this chaotic effects, really about the best we get is 6 to 8 hours. Not very much folks, and that's because there's a lot of alerts. Are chaotic dynamics, if we take a look at our planetary type system. Look at the planets going around, the currents, we have other planets tugging on them, we're going to have expansion effects, and so rather than being surprised if we hear that such and such planet Pluto. Well that's no longer a planet, well let's say Pluto or we find out that Saturn, Uranus, or one of these other have chaotic type, small chaotic type behaviour. We should not be surprised, in fact it's the other way around because of the recurrurence and expansion effects. We should expect to have chaotic type of things, so look at the Stock Market. We surely have recurrurence going back and forth. We have expansion, due to inflation, due to maybe a heating up of one thing or another. And so in economics, we must expect chaotic type dynamics to occur, rather than rare folks. My point is before I get into the next issue. My point Is that you should expect chaotic dynamics to be in your daily life. You just have to know where to look for it.