 Thank you, Professor Holti. Professor Stephen Smale's contributions to topology, geometry, dynamical systems, numerical analysis, and computational complexity theory are fundamental and extensive. To avoid stultification, I'll mention only a small sample of these accomplishments. In his PhD thesis in 1957 at the University of Michigan, among many other results, he proved that it is possible to turn a sphere inside out smoothly with no cuts or wrinkles. In 1960, while he was visiting the Institute of Pure and Applied Mathematics in Rio de Janeiro, he carried out some of his most renowned work. He devised the well-known Smale horseshoe, pictures of the first iteration of which appear in your handout, brochure, and he settled for all dimensions greater than or equal to five, the famous Pancaré conjecture, a fundamental conjecture which had been bedeviling topologists since Pancaré first stated it in 1906. This work led to his winning the Fields Medal, the highest award in mathematics. Professor Smale has written of that idyllic time, quote, my best known work was done on the beaches of Rio de Janeiro. When one considers this setting, this work must have required not only Newton quality insight, but also formidable powers of concentration. The Fields Medal was awarded to him at the International Congress of Mathematicians in Moscow in 1966. The Vietnam War was dominating the world scene. Upon receiving this award, when the Soviet press interviewed him, Professor Smale castigated the US for its role in the war, much to the pleasure of the Soviet newspaper men. Their pleasure evaporated quickly, though, when Professor Smale then leveled equally scathing criticism against the Soviet Union for its suppression of the Hungarian uprising a decade earlier. This outspokenness, together with his work in the free speech movement at Berkeley, did not pass unnoticed. As far as I can determine, Professor Smale is the only recent Nobel conference speaker to have been issued a subpoena by the House on American Activities Committee. Eventually, he was fully vindicated. In recent years, between theorems, Professor Smale has traveled the globe, figuratively turning it inside out, searching for collecting and photographing exotic, beautiful minerals. An exhibition of his work is currently on display at an art gallery in Berkeley. He has also, the last couple of years, taken time out to experience firsthand turbulent motion of fluids by sailing a 43-foot yacht to French Polynesia and back with a total crew of only two other mathematicians. Two years ago, he and his wife spent two weeks in a remote area of Kenya with their daughter. She's studying hyenas, while Professor Smale would tease herds of wild elephants by darting among them in his jeep. We are grateful that that jeep didn't fail him and Professor Smale is here today to speak about the role of mathematics in chaos. Professor Smale. Thanks very much, Ron. It's a great pleasure to be here at this Nobel conference testing, okay? So today, I'm mainly going to speak on the subject of what is chaos, maybe a little kind of a historical aside. But let me start with some, my own personal perspectives on the questions of just how can mathematicians contribute to science? How does mathematics contribute to science? Test this out. Is that very visible? In fact, what I want to do is to talk about these, perhaps three different rather arbitrary ways in which mathematics contributes to science. My own viewpoint, perhaps very biased, is that I have them listed in order of importance. And moreover, I was running up my paper for this, a proceeding to this conference, I wrote a few paragraphs before I got stalled. And that deals with the first, how mathematicians can make a contribution to science through discovery, as I say it. And so let me read a few of these, couple of these paragraphs. Since the mathematical universe of the mathematician is much larger than that of the physicist, mathematicians are able to go beyond existing frameworks and see geometrical or analytical structures unavailable to the physicist. Instead of using the particular equations used previously to describe reality, the mathematician has at his disposal an unused world of differential equations to be studied with no a priori constraints. New scientific phenomena, new discoveries may thus be generated. Understanding of the present knowledge may be deepened via the corresponding deductions. As an example, those who anticipated general relativity, it was the mathematicians who first saw that geometry did not have to be Euclidean, that there could exist in principal different possible worlds. And also the language tensors and all the tools of non-Euclidean geometries were developed before Einstein and those discoveries were crucial for Einstein's great achievements. Another example of this is found in the history of chaos. Physicists and also mathematicians, many mathematicians as well, were slow to recognize chaotic phenomena because they were oriented towards solving particular equations and analyzing particular solutions. But even a hundred years ago, Poincare had seen the possibility of homo-clinic points by transcending that methodology. Let's see, I have another slide. I was a little depressed this morning though, because I was going to talk here today about trying to explain what homo-clinic points are. That's, in fact, kind of the main goal of this talk. I was talking to Heinz-Ido Peichem at breakfast and he said it took him two months to understand homo-clinic points. And so I don't know, maybe I can give some kind of a glimmer of the idea at least today. So the more recent development of the theory of homo-clinic points after Poincare, much after occurred when mathematicians took the bolder step of looking at the completely arbitrary differential equation no longer tied to the physical world. This gave the freedom to create geometric constructions unhampered by specifics. In this broader picture, the centrality of homo-clinic phenomena became clear and an appropriate analysis carried out. That analysis was then applied to the traditional equations of engineering systems. Well, to go on a little bit, there's another example of mathematicians contribution via discovery, I think is the Ruel Totten's theory of turbulence. The second way that mathematicians have contributed to science is a kind of consolidation of physics. I sort of identify physics and science here to crudely, perhaps. Mathematicians have done an important contribution to physics by making the physical theories which are oftentimes too fleeting. They come and go too much, perhaps. But sometimes the mathematicians can help nail down those physical theories to give them a great endurance. I'm thinking, first of all, the mathematical foundations of quantum mechanics. Where we have spaces of states, Hilbert spaces with their operators, usually given by differential equations, and that formalism has made a kind of consistency proof of the theory of quantum mechanics. At least, well, I think, not only just for mathematicians, but for scientists in general, that kind of solidification or consolidation has really been important to make quantum mechanics as solid as it is today. Another example is the Landford proof of the Feigenbaum Conjecture. I think the main things were the discoveries of Feigenbaum, but they were consolidated by giving them a mathematical proof. Another example I may mention later is the geometric Lorentz Attractor. Lorentz did all this numerical work, which was quite convincing about a strange attractor. But still, one doesn't know for sure if all that numerical simulation was correct. And to have a model given, the model is due to Bob Williams, Gunkenheimer, and Jim York, they have a mathematical model, which was completely understood, which looked very much like the numerical work of Lorentz did consolidate Lorentz's discovery. And I sometimes think that that kind of consolidation is not appreciated by maybe the public at large. There's a third way in which mathematicians contribute to science, it's a descriptive way, and I think that's important too, but perhaps less important than the first two ways, because of this aspect that, if it is just purely descriptive, then there's a kind of phenomenological limit to what is said. And I made some critiques of catastrophe theory along those lines 15 years ago. As an example, Renee Tom would say that the pocket of a kangaroo is the butterfly catastrophe, and one couldn't argue with that, but on the other hand, there's a kind of limited insight, just a purely simple description gives. It's limited. And then the question, which I hear put to me many times, how about my criticisms of catastrophe theory, does that apply to chaos and fractals? And well, maybe I won't go into that here. Okay, so then let me go to the main things I want to say, which I have to do, with trying to give my own picture of what is chaos, trying to give some kind of, I don't know, maybe a definition if you want, or an assertion, answer that people would most likely give to the question, what is chaos? Well, I think Mitch Feigenbaum did that in his talk, a dynamics with a sensitive dependence on initial conditions, and it's very hard to argue with that, but perhaps one would like a little more, and that is some quality of robustness of the dynamics. When I say robust here, I mean that those properties are preserved under some kind of very minor perturbations of the parameters of the system. For example, you had sensitive dependence, and you changed the parameters ever so slightly, that dependence would disappear. It would be hard to call that really chaotic truly. So I'd like to add that the dynamics be robust. And what I want to do now is propose a little different answer to the question, an answer with a mechanism, an answer that chaos is so and so, give some kind of deeper picture of what is going on, and explain why this definition generates the sensitive dependence and also the robustness. So this assertion here, definition, will have these properties up here, the sensitive dependence, the robustness, and it will also have a mechanism involved. And so this statement is, chaos is a dynamics in which there is a homo-clinic point. To be a little more technical, I should say a transverse homo-clinic point. So I would like to explain that statement and defend it. First of all, let me say what a homo-clinic point is on a very non-mathematical level, non-technical, non-symbolic level. A homo-clinic point, I should say a bit of history here. I've already mentioned Poincaré, and as far as I know, Poincaré was the first to really maybe even see homo-clinic points at all. Probably was. A homo-clinic point, I think the terminology is due to Poincaré also, is a state of a system, which if you go forward in time, take that state, go forward in time, you reach an equilibrium, and if you go backwards in time, you reach the same equilibrium. You could replace more generally the notion of equilibrium with a periodic solution. So this seems like a fairly harmless kind of a phenomena. Just a state which in forward in time goes to the same place where it went backwards in time. This seems really an innocuous kind of statement. But I'll try to communicate how this is such a powerful idea that it really can be identified with the notion of chaos in some reasonable sense. When I say this is a definition of chaos as the existence of a homo-clinic point, I'm not saying it is the definition of chaos, because I think there could be many definitions for chaos as there is for most things. And I'm just proposing one definition which I like a lot. The definition of chaos is a dynamics which possesses a homo-clinic point. This is a true statement, but I want to be a little more mathematical about this. Give a little more, slightly more technical definition of homo-clinic point and eventually construct them and try to see how they work. Let me start at the beginning. I want to just say even what is a dynamics? For our purposes, it's going to be a special case, but just about everything I say generalizes completely. The arbitrary number of variables, the continuous time, just a very general setting. But the dynamics I'm going to use to illustrate everything is a dynamics in the plane. So I'm thinking now of a state in the plane, which our mathematicians say R2. It's the Cartesian plane if you want with X and Y axes. I'm going to take time to be discrete and reversible. So time is given with values 0 for the present, 1 is the next, 1 unit of time later, and so on. Minus 1 will be 1 unit of time ago. And I will write for a state Z I will write Z sub t to be the state which satisfies the initial condition. At time 0 Z sub t was the original Z. But at Z2 then Z2 will be that state 2 units of time later. This is very elementary and known to many of you, I'm sure. So we can write the orbit of Z, the trajectory of Z that state throughout time as these dots, Z minus 1, Z0, Z1, Z2, Z3, and so on. That will be the sequence of states which are generated by the initial state Z at time 0. Then from the hypothesis that I've said, we can express the next state by a transformation. So I write capital T as a transformation of the plane. And that gives us the value of the next state. So I put any state in our equation here and I will produce the next state by this transformation, invertible transformation of the plane. For many of you, this might be a little tricky, but I'll give an example. Which is going to be a simple example, but if we really understand that example we'll be on our way to understanding homo-clinic points. So the example I'm giving is not chaotic. It's a linear dynamics. And we have this picture. I hope that's visible. It's not too visible to me, but it's okay. It should be higher. Oh, okay, there's a little bit on the bottom there, isn't there? Okay. So this is an example of a transformation of the plane. I better move it down a bit. Yeah, there's the transformation T in this case, given in terms of Cartesian coordinates X and Y. So if we take a point, for example Z0 in the plane. Z0 will be a state. We want to see where Z1 is a unit of time later. We apply this little tiny formula. We take the X coordinate, double it. Take the Y coordinate, take half of that. So we move down here by half and out here by two and we get this point Z1. So this dynamics takes this state to that state in one unit of time. And if we apply the dynamics again, we move down by half and over by two and we get a point somewhere over here. So points are moving like these arrows direct us. If we apply it again, then we go out here somewhere. So points like this are going to be moving along in a hyperbolic-like shape under time. I think this example is even though it's very simple, non-chaotic linear. If we understand the dynamics of this example, we're already on our way to seeing something about homo-clinic points. No homo-clinic points here. So let's look at a few other points. See what happens to some other points as time progresses. If we take a point over here, again we double down here and move over. So a point here is going to move like this and again like that. You have this kind of mathematicians call a saddle kind of structure to this point. Zero is constant. Zero doesn't change in time. Zero is an equilibrium of the system. If we take a point on the axis here, on the y-axis so that x is zero then each time we apply that dynamics we take half of it. So it will go down here, here, here approaching the equilibrium as time goes to infinity. If it lies on that axis the same if we are in the negative y-axis the point here will be halved and go up here. So we have this phenomena of the set of points which move into the equilibrium are precisely the y-axis. We'll see a little more generally that's called the stable manifold. And this is one of the most important things I think that modern dynamical systems has given us is this focus on the stable manifolds of an equilibrium. The set of points which are attracted to it. In this picture here, the set of points which are attracted to the equilibrium are precisely that one dimensional straight line. The other points are moving away. If a point is not on that y-axis it's going to be moving away from the y-axis moving away from our equilibrium at zero. And we have some points which as time goes to negative infinity do the same thing that we call the unstable manifold, WU. The set of points which move to the equilibrium as time goes to minus infinity. So we have this kind of structure of our linear transformation. The two axes, the x-axis and the y-axis which give us the set of points which are asymptotic to the equilibrium. Asymptotic to the equilibrium as time goes to plus or minus infinity. I've given you some idea of this particular example because that's, as I say, going to be quite important for us. Then as I said before, if we have general dynamics, the equilibrium is a fixed point. It's a state which does not change under time. It's given by the equation tfp equals p. This means for all time that state remains stationary. An equilibrium is a stationary state. And we can define for any equilibrium the stable manifold Ws of p what I call this important development is simply the set of x which is the property that the orbit tends to p as t goes to infinity. And the unstable manifold, same thing as t goes to minus infinity. So we have this definition then which works very generally for dynamics and is one of the most important global aspects of dynamical systems which I think was an important thing in beginning to understand chaos. The use systematic use of these sets which are asymptotic to an equilibrium. And it's natural if you want to see what happens to a state in the long run you want to see where it goes. And that's exactly what's happening here. You're looking at the set of points which tend towards the equilibrium. Now I want to go beyond that linear example by a small change. So what happens if we perturb the linear example a little bit? Tell me if I'm off the picture sometimes. I guess that catches the main thing. So what I've done here is suppose we have some kind of change in the parameters of the dynamics. Not very big but just a small change in the parameters of the dynamical system which was given on the previous slide to make it nonlinear. And so what has happened now is that that set of points which are asymptotic to the equilibrium P, we still have an equilibrium P, the set of points will no longer be a straight line but a curved line like this. So all these points are going to be marching down here under time. These points also are going to be marching towards P under time. The points in W of P that I've marked here are going to be marching away from P along, precisely along this curved W of P. And other points are going to look much like in the picture of the linear case Z0 will map to here like this Z1 over here. So these other points are going to be marching along here until they get to the influence of this set and then they're going to be marching out. So we have this picture that I should see if I lose it. Yeah, in this picture here we already see the phenomena which is crucial for chaos. We have a strong contraction of points in the direction in the y direction in the vertical direction points are being shrunk because of this half in the formula. These points are being shrunk. Points in the x direction are being expanded by the factor 2 points here are being pushed apart. So we have the contraction along here and the expansion along here. And that's I think one of the crucial features signatures of chaos. And that situation prevails over here in this picture which is no longer linear. Points over here are going to be moving along here until they get down close to this and then moving out here like this. So you have these lines here like this. When I draw a curve like that I do not have a differential equation of the flow. Just that points are going to be marching along in the direction of that arrow. Under our discrete time of this dynamics. Okay, so we've perturbed our linear situation a little bit. Still no homo-connect points, no chaos. In the picture the dynamics here is very similar to that of our linear example. But now we want to do it a little bit further. I'm going to make a little further perturbation. So now we have something which might be called strongly non-linear or maybe all together a big perturbation. It's a perturbation in such a way that you have a crossing now of these curves. The stable and unstable sets are not a false now. We suppose a cross at a point Q. Okay, well, what does that mean? Remember I said a homo-clinic point was a point which tended to equilibrium as time goes to infinity and as time goes to minus infinity. So that means precisely an intersection of this curve with that curve. So from what I said so far we can see a homo-clinic point is precisely an intersection Q then belonging to WS intersection WU. And it's not the equilibrium itself. That's a homo-clinic point. The one that tends to equilibrium forward in time and backward in time. Okay, and the transverse, the technical condition means that the crossing here is not tangential. A minor technical condition. Okay, so here we've been a little bit more geometric about our definition of homo-clinic points. We see the kind of situation where they might occur. And a lot needs to be said. And especially when we go to the question, how does this give us chaos? How is this going to say anything about chaos simply just a crossing like that? It looks harmless enough. It doesn't look like it should produce chaotic dynamics. Okay, but let me then try to look at some of the consequences of just simply the existence of homo-clinic point forces. Some of the consequences. Okay. Well, the points on this curve are invariant. So if I apply the dynamics to Q, it will stay on this curve. It will also stay on the curve, which goes in negative time to P. So a homo-clinic point is preserved under the dynamics. Okay. So if I draw it this way, the transformation is going to have to take Q to a point from here Q, say Q1 which is also a homo-clinic point. So Q gets mapped under the dynamics into a point which also lies on this curve. Okay. And since we have to preserve this crossing, it's going to look a little bit like this. So the transformation is going to take these points into these points. How can that be? Well, the only way for that to happen is for this curve to double around like this. So if we have a homo-clinic point, it forces this bending around. Just the existence of a homo-clinic point forces this picture. But moreover, if we go backwards in time, this point Q must also go back here to a point Q-1 one unit of time later on this curve. The same kind of picture like this. So we've seen just from the existence of the homo-clinic point, we began to see a little bit of a things that we could say is getting a little complicated which is true enough. I drew, I have another little better picture of the same thing here. So this is the picture that we've deduced simply by the existence of Q. Okay. By the same argument again, this point is a homo-clinic point now, so its image will have to be down here something of the same type, Q-2. And so we can argue that this must curve in here like this. And now we see as we're getting under the influence of this curve, it's going to be pulled out like this maybe, or perhaps pulled out farther and come back further. So this situation forces a bigger loop down here. The loop is getting bigger because of the stretching along W of P. So we have this stretching effect. And if we do it again, it gets stretched a little more and it's going to come into here like this. So we, in fact, we get more homo-clinic points crossing here. We've created more of them. Q-2 is a time where we get completely new homo-clinic points forced by this picture. Well, that would make the same argument on these. Poincare wrote that I will not try to draw the figure implied by the existence of a homo-clinic point and I think maybe I'd better stop. But here's the same thing, maybe a little cleaner. You get the new homo-clinic point there where the same analysis applies to that. There's many more new ones and pretty soon you get confused. I hope I've communicated a little bit of why this kind of complication follows from our original assertion of just the existence of the homo-clinic point. But how to deal with this complication? One can't quite say this is chaos so far. This is just a mess. But maybe it's suggested that this could be chaos. And in fact, as I asserted, the existence of the homo-clinic point I need to do some more justification. So what we'll do is we'll take an original picture. I have that very handy here. Yeah. The original picture is messed up. But the original picture, remember, of a homo-clinic point looks something like this. Here was our P and here is our Q. So remember now we're shrinking along this arrow. We're shrinking in and we're expanding along this arrow from P to Q this way, shrinking this way. So what I'll do is to take a little rectangle which covers that area. I take a little rectangle like that. I take a little rectangle like that and that's going to be shrunk in this way and pushed out this way. So if I take that rectangle on the picture, if we look at it sufficiently, number of units of time later is going to be pushed out along here. So we have this picture. This rectangle is being shrunk in this direction and expanded in this direction to give us this new rectangle which is just placed over it with our point P in here. Okay, so let's try to straighten that out. This then is just going to motivate this picture which is drawn in our program even, the the harsh shoe. So if we just take this and we make one of these rectangles into a square, we get this picture. So we keep the same general configuration but just put it into a little more place where we can analyze it. So now we have the square here, well I messed up the square but it's almost the square and by the construction it gets shrunk this way and expanded this way. It's shrunk in this way and expanded out in this way. So it gives us a long thin rectangle and bend it around and then place it back here and we get this. So what is going on here is the following. We have the square is mapped onto this bent rectangle. In such a way it takes this line into a prime d prime and it gets bent around and if you think through then this gets turned around so that bc gets upside down and it gets c prime d prime. So the idea is that c goes to c prime and so this rectangle gets mapped into the harsh shoe and if we are a little careful we can see that if we have a homo clinic point we must have this picture in a very very general statement that works in arbitrary dimensions any kind of topological situation any kind of dynamics. You will get this picture from or a slight generalization of this picture from the existence of a homo clinic point. Now let's just forget about the homo clinic point which was quite confusing and let's just look at the dynamics of this picture in its own right. Forget what happened before with the dynamics given by just these boundary conditions. What does that dynamic look like and what can one say about it? Well let me just start this analysis and indicate how it goes. We look first of all at the set of points which go into this dotted rectangle and if you follow through on this picture here you can see the set of points which go into a piece like this there is a little strip here and that will be a strip like that here. So the set of points which are transformed into this long horizontal rectangle come from this vertical thin rectangle. So this gets mapped into here. Similarly this rectangle, this long thin one gets mapped into this one down here and one can analyze a set of all points which persist in this square all time forward and backwards. They will be labeled by what time they're in this rectangle or this rectangle. So we just mark zero here, one here. Say zero here and one here. And we label a point by the rectangle that lies at time p. So that gives us for each time a zero or one and that way to each orbit of this dynamics we have a sequence of zeroes and ones and one can show that any sequence of zeroes and ones can be produced this way so that the dynamics here of all the points which persist under all time forward and backwards all those points correspond to sequences of zeroes and ones with a decimal point and the dynamics itself corresponds to moving the point one place so the dynamics is now just this. Okay, what's the point of all this? Well, what that does is to reduce the qualitative dynamical picture to this combinatorial picture. And in that picture and this picture is zeroes and ones one can that had already been done for 50 years or so answer all the dynamo almost all the dynamical questions can be answered by combinatorial playing around with these sequences of zeroes and ones. One can say something about the periodic points being close to the homoclinic points, one can read off the homoclinic points from here and so one can interpret all this now to points in here so the homoclinic points will now correspond to a certain subset of these sequences of zeroes and ones so some of the consequences we obtain from all this so first of all one can take the following and one can just define a map which looks like a horseshoe, it's easy to do one just takes that square thin it out and do that procedure I mentioned before to actually describe a dynamics which looks like this so that kind of dynamics does exist and we get immediately homoclinic points do exist something that for example Pochere was never able to do I don't know if he ever tried in fact in those days I don't think they tried too hard to prove theorems but in any case it would have been difficult for Pochere to try to prove the existence of homoclinic points, he visioned them he could see them, he believed in them but it was hard for him to be able to really assert their existence in any kind of rigor or any kind of constructive way that he didn't have a way to construct homoclinic points so we obtained by going one way in this picture starting with a horseshoe we get the existence of homoclinic points then we can also analyze the homoclinic points as I said we conclude lots of these questions about nearby periodic points sensitive dependence now follows from the picture if you look at it very carefully you see the sensitive dependence on initial conditions and that goes back to these contractions and expansions from the contractions and expansions you see all the points which persists in that square do have sensitive dependence moreover we get the robustness of the same that picture is robust so that means if a dynamics has that it's going to persist under any kind of any kind of perturbations let me give a little of the historical picture here Jim Glick in his book Chaos I think it's a nice accurate account of a the horseshoe let me just detail some aspects of his account so that's in the background here we should keep in mind that homoclinic points had been to a great extent lost Ponker had found them and everything and Burkoff had done some nice things with him but when I was working in differential equations in 1959 the whole community was unaware of homoclinic points everybody I talked to seemed to be unaware of the existence of homoclinic points and I talking to people even today they said that it's a pretty prevalent phenomena that nobody knew what homoclinic points was were in 1959 I shouldn't say everybody but essentially they weren't part of the culture current right in Lillewood great English mathematicians during the war were given some problems on understanding the equations of electrical circuits some war work they were doing and they had maybe a 100 or 200 page article which give a lot of analysis of this very simple differential equations two variables forcing a lot more time where they notice a lot of very interesting phenomena complicated phenomena like an infinite number of periodic solutions and they detailed this out in a great but almost unreadable paper of this long long paper which I don't think people read except for Norman Levenson actually Mary Cartwright is still alive and I visited her at I guess it was Gertin College Oxford or Cambridge probably Cambridge Norman Levenson an American simplified some of the differential equations in this paper of Cartwright and Lillewood and pulled out some of that exotic behavior I was on the beaches of Rio de Janeiro in 1960 besides swimming I was doing some mathematics I was there for six months and at that time I received a letter from Norman Levenson which said some conjecture I had made was wrong and he gave us an example his paper I spent a lot of time trying to understand his paper in a more geometric way at that time that I gave this horseshoe picture and in fact so that was in 1960 and the mathematics of time by Springer it's a book of my 1980 there I have an essay on how I got started in dynamical systems which gives a somewhat detailed account of all these things so summarizing a little bit we have this little diagram chaos I've asserted and given some kind of justification one has to give a lot more can be identified with homo clinic points I think it's fair to say there are some borderline cases which one might call chaos homo clinic some very borderline cases but I think essentially it's a it's not bad to make this identification homo clinic points can mathematically rigorously be identified with a horseshoe generally but that's not really the whole story because what I've talked about is a kind of a I've used chaos in a very general sense one can talk about something which one might call full chaos full chaos means that chaotic behavior for essentially all states states with probability one and in the picture I've described the homo clinic point in the horseshoe it only guarantees the chaotic behavior of some set of points except for a little while let me go back to this I don't know if I have it very well here but see in this the horseshoe if we look at a set of points which really stay in this square here we get something like a counter set of points and so what happens at any point which tends to the counter set that will be truly chaotic so that's a set of measure zero dimension one at least or more so the probability one we don't get a set of full measure which is chaotic now I think a lot of research has vindicated this lately that points will tend to stay in this chaotic region much more long than one would have a reason to suspect so the points can march in here even though they're not on this counter set but they will stay here for a long long time acting chaotically and eventually may go away to a stable solution so you have this transition transitory chaos transient chaos which works for many many points not just a thin set of points but many many points but still one would like to talk about something where points with probability one are chaotic and so that's a question of looking at an attractor a chaotic attractor or a strange attractor or a chaotic attractor so full chaos means that states with full probability will behave chaotically and I have this picture full chaos implies chaos the chaos itself has homoclinic points so certainly full chaos will also have homoclinic points so we have this idea now of an attracting set like this so that all points nearby will tend to that chaotic attractor so you have all these points a set of full measure going asymptotically to this chaotic set which will then give us a full sense of chaos a full definition of chaos and I think remains a major questions in science as to whether the chaotic phenomena we see can be explained or not by transient chaos or does it need full chaos to explain them I think those kind of questions are not really understood an example of a chaotic attractor is the Lorentz attractor and in that situation surely states with full probability will act chaotically for all time so let me close now with a couple of problems for me a major problem of science of chaos turbulence 2 is to identify chaotic attractors in physical systems if we do that then we can say that we really have proved the existence in that physical system of full chaos that has not been done ever to my satisfaction in general I think it's hard to justify that it's ever been done to strongly identify a really solidly identify a chaotic attractor in a physical system one can do that in mathematically constructed systems in geometrically constructed systems one can do that very easily and there's a whole slew of lots of examples where one has full chaos but in the systems from physics it seems to be a very very difficult system very very difficult problem and I'm not asking necessarily for a rigorous proof but just some kind of solid proof that really understands the numerics which proves, gives numerical simulation or the mathematics of the equations some even less than rigorous level try to make this identification I think I won't say the major problem of chaos but it certainly to me is a major problem of chaos to make that identification and it could turn out to be the situation that really gives some understanding of for example this could be done with the equations of fluid motion if that could be done with equations of fluid motion under realistic circumstances this kind of result could give us an understanding of turbulence so particular examples of this problem example here equations of Lorentz we know numerically what they look like they look like they have a strange attractor and that simulation has inspired this work on the geometric Lorentz attractor I've mentioned which is understood completely but one has not connected this up with this so mathematically one has not proved the existence of any kind of an attractor for the Lorentz equations those equations which help them inspire this whole movement towards chaos theory in that situation that most possible situation of chaos one believes that there is a strange attractor but it's not proved at all you can say it's an academic question and some physicists would say it's an academic question but I don't think so I think there's some missing link here and until one really understands this question it may well be there's something conceptual that's very important that we don't understand about chaos until this question is understood this is a very special case of identifying equations in physics with a strange attractor of course one could say the Lorentz equations aren't exactly physical but at least they had some kind of original motivation in the equations of weather and the final one question well I'm into question these days I gave a talk in May on nonlinear systems in my paper which is finished now Dynamics Retrospective Great Problems, Attempts that Failed so I talk about all the things I couldn't do for many years and as many of you know I haven't worked too much in dynamics or chaos since the 60's and so as I say the reader should consider this note as reflecting a voice out of the past and I list ten problems which I consider great problems of dynamics problems that I find very hard couldn't do, two of them are these problems do the Navier Stokes Equations possess a chaotic attractor I pose a very particular case of this problem and the Lorentz problem in any case let me stop there thanks very much