[MUSIC] What I'm going to do now is give you some examples. And give you one particular example where were going to understand what causes chaotic behavior and why you, you have to expect it almost everywhere, including, your morning coffee, or anywhere else. I'll explain what that means. To do this I'm going to look at an algorithm. Which is deterministic now. An algorithm all to determine a zero of a function. First question is the WGAD question. W G A D. Who gives a darn, okay? Who gives a darn about finding a zero of a function? You should. Let's take a look at from engineering sciences. In engineering sciences, what we're doing is we're building a bridge and we have the tension between the two different parts. And what we're trying to do is find where the tension on this side, and the tension on that side, it's in an equilibrium where it's going to stand still. And boy, when you go across that bridge, you want to make sure that those zero equations are being satisfied. Same thing in economics. Remember Adam Smith and the invisible hand? The invisible hand where the market dynamics is going to move it to where supply equals demand. Supply equals demand, that means supply minus demand is equal to zero. To understand part of the economics I need to find the zero of the excess demand function. Here's another one. Let's try to find the maximum or minimum of something. Suppose I through something in the air and I want to find the maximum height. I have the equation for where it is at each instance of time, so I just want to know what time do I plug into that equation to find the maximum height. Well, look, if I throw it in the air, it's moving up, it's slowing down as it gets up, and it gets over to a point where it's equal to zero. The velocity is equal to zero when it comes down again. So if I can find the zero of that velocity, then what can I do, is I can plug that time in into the function and I know the maximum height. This is precisely the Calculus II which is used if you determine maximum profits, maximum, minimum, whatever you would like. It is defined as 0 of the rate of change. All right. So let's now go to the question and finding the zero of a function. Now what I have right here is I have a function. You see it's curving down. I have absolutely no idea where the zero is. Oh, on the picture I know where it is. But how am I going to find that mathematically? So the approach is, and the approach was defined by Newton. And what Newton said is, let's make a guess. So we guess the point x1. So I choose x1 and watch. Whoops, nope. It's not a zero. It's not a zero so I need to find a second guess. Now what am I going to do for that second guess? Well, as Newton recognized, the function itself is far too complicated to analyze, so when you said let's replace the function with a straight line that comes down here. A straight line that's easy to analyze, that's easy to determine where the next zero is going to be. We learned that in the first course of algebra, so we have the next guess. Notice that next guess is closer It's closer to the zero than the first, but we can improve on that. So we move down, oh not a zero. Replace the factual curve with a straight line. That shoots up and gives us point X3. And notice X 1 was, well, it was a guess, but it was off. X 2 is closer, X 3 is even closer, you know what's going to happen. X 4 is closer, 5 is closer, 6 is closer, and it's going to converge, and actually it converges rapidly to the 0 of the function. Newton's method is a way which we commonly use to find zeroes of functions. It's a powerful approach. All right, time to look at and find if anything can go wrong, whether Newton's method allows us to have chaotic dynamics. And just the fact that I, raising this question, you know it can. So let's look at this more general function that I have here. Take a look. It goes up and down, and up and down, and up and down. We know that there's going to be regions, Newton taught us that, there's going to be regions around where that line crosses the x axis where there's a zero. We know that there's going to be regions which are well behaved. What I want you to do is to perhaps run pause, and I want you to think of what can go wrong with this particular function if I using this method. One thing that goes wrong of course are these points right here where you need to go down and what you're going to have this horizontal line. A horizontal line, the next iterate is going to be off at infinity. That's going to be a little difficult, so that's a bad point. And if you notice on this drawing we have 1, 2, 3, 4 bad points. So, what I have here are straight lines coming down, you can divide into regions that I'm going to imaginatively call A B C, so those are the regions. Now in understanding and in analyzing Newton's method prior to, I would say about 1980, the approach was. You take a point, follow its iterate, by all these careful hard computations. Find the next iterate, find the next iterate, find the next iterate. Where things had go bad. And what did they discover? Well, they worked their fingers to the bone, and about all they got were bony fingers. It was one of those type of situations where you can work hard and get little. Let me tell you what they did, and then tell you what we do now. They chose a point, x 1. They found the iterate x 2, the iterate x 3, etcetera. So what happens is maybe the point x one was in region A, maybe the next iterate was in region C, the next one was in region B. So it bounced around and then they tried to analyze, what kind of question am I going to ask? They had difficulties in computing this. Let's do it the other way around. Rather than starting with a point and analyzing its future, let's pose a future and try to determine whether or not that future can be realized. This is an area called symbolic dynamics. So to do this, [COUGH] I want you, I'm going to carry out one case. But I want each of you that are watching this tape. You write down your own future, of where you want the iterates to go. And to follow what I did, and to show exactly what happens. The future I'm going to start with is the following. I'm going to say lets start with a point in region A, then let's find, say that it's going to jump to region C. After that it's going to jump to region B. Eh, you tell me which one. Maybe after region B maybe it'll jump to to region A for, what, 50 times, 1000 times? I don't care. And then after that jump to region B then C then A then C then B. I don't care. You write down any future you want. And the surprising fact. Is that there does exist an initial point which has that future. I'm going to show you how to find that point. And as I stated, I want you to write down your own futures, different than the one I write down. I want you to write it down, and I want you to try to compute, carry out what I'm showing you ,and to compute where that initial point would be. Well, [COUGH], what is it that I want to do? I want to start in region A, I want to jump to region C. Alright, that's fine. That's dandy. How am I going to do that? To do so let me confess to a wasted youth too much pool. Not very good at it, but too much pool everything else. And so let's go to find how you would do this with pool. Suppose the goal is to hit the cue ball, to hit the 1 ball, to hit the 6 ball, to hit the. You write down the future that you want to have. Just like I wrote down the future of where I want Newton's iterates to go, you want the future, you're playing pool, to go in according to your plans. Professional player goes up there, looks that over, and [SOUND] boom! And away it goes. Not me. I'm lousy. So what I have to do is what all of you have to do. First I take that cue ball, and I look at that cue ball, and I say, where do I hit that cue ball? So that I'll hit that one ball. So what I do is, I look at that edge of the one ball, and I come back to say, oh, if I hit the cue ball here, it's going to hit that edge. Then I look at the other edge of the cue ball and I bring it down on, say, here, and so I have a target region. So I have a target region on my cue ball which says that if I hit that target region anywhere, It's going to hit the one ball. So, lets do the same thing for Newton's method. Let's go back and take a look. I want to find where in region one, what's the target region in region A where it's going to hit somewhere, I don't know where, but somewhere in region C. Well, that's not too hard to do. What I do is I just take a look just like I did with Q for the cue ball in ball one. I find here is the extreme on region C, the extreme right point, and I say, I draw, put in a straight line so it hits that extreme point, and I go back and say oh, if I'm right here I'll reach an A, it's going to whammo, it's going to hit that far extreme point to reach in C. Then I'll look at the other boundary. The closer one. I want to hit that. And how am I going to do that? Well, I just draw that straight line backwards and say, whoa, if I'm here, it's going to hit there. So what is it that I have? Just as in the cue ball story, I know where. I have a very small region, notice how small it is. I have a very small region. That if I start in that small region Newton's method is going to expand it and it's going to cover region C. Got the first step. Problem is to get to the next step. So as far as in pool, I got the one ball, but I don't just [LAUGH] want to hit the one ball. I want the one ball to hit the six ball. So what do I do? I take the cue ball, walk over, get behind that one ball seeing where do I hit the one ball so that it'll hit the six ball? Do the same thing. I find well let see if I hit the one ball here it's going to hit the edge of the six ball. Hit the one ball here it's going to hit the other edge there. So I got a target reaching on the one ball. So I do the same thing in Newton's method. Newton's method I'm going to now say. Where does the target reach? It reaches C. The target Region and the Region C, so that I'm going to hit Region B. So I take the extreme point on the B region. One farthest to the left. And I draw this fine, what is this straight line that'll hit the curve over there, over the B? And bring it down and say, ooh. If I hit here, this method is going to take it to that extreme left point. Then I take it to the extreme point of region B and I do the same thing. And I have a target region in C that if you're in C, you will then move over and you will hit all of B, somewhere in B. Where? I'm not sure, but it'll be in B. All right. Let's go back to pool. I know where to hit the one ball. I'm not hitting the 1 ball, I'm hitting the cue ball, so I have to go back to the cue ball and say okay where's that refined region? I got this wide region here if I hit the cue ball it'll hit the 1 ball, now I want a refinement so it doesn't hit the 1 ball but it hit's the 1 ball where I want it to hit it. So I refine it. And what happens is it hits there and it'll hit the six ball. And away we go. Precisely the same thing in Newton's method. What I have now is I have this small region, in the C region, refined region, that if I hit there it's going to hit the B region. So then what I do is pull that region back to the A region. Instead of all of C, remember it had that little interval in A that if go in A I'm going to get all of C. I don't want all of C, I want to hit that small region C. And so I have this incredibly small little region. Where if I hit in this region it's going to get into that small region in C and C is going to hit anywhere you want in B. What happens next? You tell me. You just continue this. You tell me what you want to be the next future. What you want to happen after B, and that's fine. You know, just from what I stated I'm going to do the same thing. I'm going to find a target region on the six ball so it'll hit the nine ball and then I refine region on the one ball and then refine region on the cue ball. I'm going to find, you tell me what you want to happen next. I'm going to find the refined region on the B region so that it'll hit where you want. That refinement on the C, that refinement back on the A. Oh. Look at that incredibly small region that we have in the A region that will have all of that future. Notice what's going to happen in that incredibly small region. The first three steps are, start an A Goes to C, goes to B, and what after that? You decide. It could end up being as random as you would like after that third step. So all of these regions are very very close together and they can have incredibly random appearing future. That folks is chaos. That's chaotic finance. And so this type of chaotic dynamics that I"m showing with Newton's method is really captures the explanation of what happens whenever we see a chaotic. Dynamics.