 Dr. Feigenbaum received a bachelor's degree in electrical engineering from the City College of the City University of New York in 1964 and the next fall attended the Massachusetts Institute of Technology as a National Science Foundation Fellow. He received a PhD from MIT in 1970 in theoretical high energy physics then he became a research associate at first at Cornell University and later at Virginia Polytechnic Institute. During his eight years at Los Alamos National Laboratory he began his pioneering work in chaotic dynamical systems spurred on by attempts to understand turbulence. His work at Los Alamos earned him a fellowship and the Distinguished Performance Award. In 1982 he returned to Cornell as a professor of physics where he continued his research on chaos. In 1987 he joined the faculty at Rockefeller University as the first Toyota professor. Professor Feigenbaum has received many other honors for his work including the Wolf Foundation Prize in Physics, the MacArthur Foundation Award, and the E.O. Lawrence Memorial Award. He is a fellow of the American Physical Society and a member of the American Academy of Arts and Sciences and the National Academy of Science. He is a member of the editorial boards of the Journal of Mathematical Physics and advances in applied mathematics. Dr. Feigenbaum's work in chaos has been instrumental in establishing this new science. He has authored at least 20 articles in the area of chaos, non-linear dynamics, and turbulence. Until the initiation of systematic studies of chaos physics had been unable to predict or describe events in nature that exhibit erratic or turbulent behavior. Dr. Feigenbaum discerned distinct patterns in the transition from orderly sequences of numbers to disorderly. He began with the realization of the futility of understanding the complexity of turbulence and proceeded to study its simplicity. In the study of simple non-linear equations he found that different equations produced the same pattern. He was able to demonstrate that these patterns possessed the mathematical property of universality. The patterns of transitional disorder were identical regardless of the physical system. Dr. Feigenbaum was one of the first to describe the period doubling route to chaos. This theory has been used to explain transitions in chemical, biological, and physical systems including dripping water faucets and non-linear electronic circuits. Dr. Feigenbaum continues to study universality and transition in chaotic systems and is presently considering the problem of the sensitive dependence on initial conditions. We are very fortunate to have Dr. Feigenbaum with us this afternoon to address the subject transition to chaos. Dr. Feigenbaum. I think following that introduction I don't have very much to talk about any longer but I'll try to fill in some of those details. Also for the first time that I've ever done so I have a written talk in front of me and I have no idea what that will do to my style and I'll probably bypass it at some point. Well the study of chaos as you might know is a study that's immersed within a rather larger study and that larger study is the study of so-called strongly non-linear systems and during the course of my talk I will return to telling you what strongly non-linear systems mean and of course to say something or other about chaos. In the discipline of physics there's a most prominent example of what serves as a strongly non-linear chaotic system if not worse than chaotic and that problem if you will an exemplar is the problem of turbulent motion in fluids. Chaos of course is not really the study of the turbulent motion of fluids. On the other hand it serves as a very powerful icon that any good physicist working in the subject will keep in the back of his mind to always remind him what are the kinds of questions he should be endeavoring to understand. As with all rather good icons one of the virtues is that you can immediately although only vaguely understand some general sort of questions you would like to discuss. You can of course do better that than that and think of some rather precise questions you would like to understand however it's important to realize that if you don't really understand what an object is what are the right things to think about about that object then you can't really figure out in advance what questions to ask as to how those different objects questions thoughts whatever should be related to one another and so while there exists a rather separate entity from science which is I suppose the entity of technology in technology the circumstances often different one is definitely faced with problems one would like to build submarines airplanes various things like that that must move within an environment that exhibits turbulent behavior and for the need however that need is perceived of constructing these objects one is then forced to offer some sort of understanding however in the discipline of a science one has a rather different obligation one doesn't necessarily only want to give answers the sort of quick fix answers in fact to the contrary one really wants to understand what are the right things that one should be doing to come to embrace a larger class larger that even than the questions that you're thinking about as to how one analytically quantitatively can describe various aspects of nature so I will mostly be talking about things from much more of a scientific perspective the fundamental questions as I perceive them are what is the nature of the objects that we're observing what are the right ways to describe them and if we know something about that then how do we formulate rather good questions to ask which have to do with relations between these things well continuing with this image of a complicated fluid motion you should think of all kinds of swirling things there are lots of parts if you look at a fluid if you look at the foam at the bottom of a waterfall you know there are lots of different parts all doing very complicated different things and you have to ask yourself how are you going to describe these innumerable numbers of parts together with a complicated motions well historically say since the plus is been a rather well understood method it's a method that any scientist any analytical scientist will turn to and the method is of course a statistical method that simply means that rather than discussing what a million different things will do one would rather at least like to know the act the answer to what will some well-chosen representative if you will averaged quantities do instead easy to offer examples of that if you observe a certain population voting or a certain population being pulled as to how it might vote then of course one thing that's rather easy to understand is a statement that 34% of the people voted for this particular person and it's probably preferable to get that answer then to have presented in front of you the list of the several millions of voters each one having voted for this one that one or the other one the same situation is of course true in various physical circumstances if I think of a gas of particles the air in this room then it's an easy question to ask an answer what is the total number of particles which is in some sense of rather measurable quantity inside the volume of this room which is quite easily measurable that is I'm interested in knowing how many particles on the average per unit volume or there that's a rather easy question to answer a very different question is to look at the Avogadro's odd number of particles 10 to the 23rd as you've heard before for each one of those particles to say at any one moment of time where is it now how quickly is it moving and in what direction that obviously is an insuperable sort of question to ask it's insuperable even to write down the answer probably that doesn't exist enough paper to write down what such a statement would mean and so often in these various pursuits of looking at complicated things one replaces the problem of asking for sort of the most information by a rather different problem of asking what are more average properties that we might talk about well that's all well and good it's certainly good if you set about and try to measure them although that can be orduous in many circumstances but as a theoretician one of course would like to know how to predict these things so while I might say it's rather easy to talk about how many particles per unit volume or there it's a different question for me to go ahead and tell you theoretically how you would come to that knowledge well as an all statistical questions there's a definite procedure there's a definite thing that you have to do if you want to be able to render such a prediction and the relevant object is called a distribution function and what it means to think for example about these problems of voters is that imagine you run the vote again and again and again and again in the end in innumerable number of times each time you run the vote you see what the answer comes out as 34 percent this time 37 percent the next time and you now count up how many times do you see each number so if you do a billion different votes and you happen to see 37 percent happening one million times that's one of the datum that you care about and you now produce the set of answers for each different outcome how often did you see the population doing that such an entity is called a distribution function if you can find that then of course you can make a lot of answers should it turn out that out of a million attempts to do the vote the answer was always 37 percent then of course the next time the population votes with rather high confidence you could say the answer will be 37 percent should it have turned out instead that you saw one million voting 37 percent 1.2 million voting 38 percent 1.1 million voting 39 percent and a rather uniform distribution then what you can say is what the average will be you can say how big a deviation from that average number you might anticipate but you certainly wouldn't want to put too much money on making a precise statement that it will vote this way so in general if you want to understand what these more average statements mean it is your job to furnish one of these distribution functions well that's well and good we now ask how theoretically do we come to do this well their things don't work quite so easily if I think about the problem of voters I don't have the vaguest idea theoretically how to predict such a thing in fact for various of of some of the sciences that we contemplate sciences of economic sciences of sociology and so forth the state of those sciences in their predictive powers aren't yet of that strength that we know how to go about and determine in advance what these distribution functions might be so from this point onwards I won't say anything further about those sort of economic sociological what have you pursuits on the other hand the circumstance in physics should be drastically different if I return to the question of what is the fluid going to do well again the fluid has so many parts to it that I would probably prefer to answer a simpler question of knowing some average information however here I want some rather specific average information fluid motions are characterized by the fact that there are sort of worlds in them there are eddies the fluid is moving around in little vortices lots of them are moving around moving past each other changing the character and so forth I could ask what would turn out to be too simple a question which is to know of each size of these eddies how often do I expect to see such and such a rate of rotation I actually would like to know even richer statements than that I would like to know at this point in space how often do I expect to see this size eddy with this value of its rotation these are rather specific questions they get to be rather detailed they almost tell you everything that's going on but I still very truly am going to average what I'm going to average over is the infinitude if you will of the innumerable little pieces that the fluid is comprised of well we are in principle and have asked me better position in the physics of these fluids because we happen to own certain microscopic laws that is we reasonably well believe that we have a very good representative set of equations the so-called Navier Stokes equations that reasonably well described what such a fluid should do but what these equations tell us is how this microscopic piece of the fluid is going to behave in the course of time so we have an assembly of virtually an infinite number of such pieces we're told for a teeny amount of time following that how each of these pieces will move from that knowledge which we believe is reasonably firm knowledge can we now go ahead and figure out what these distribution functions are that at least will allow us to answer various statistical questions well turns out that this is not either a very easy question as a matter of fact no one knows how to do it at all there's a rather happy circumstance that exists it's existed since the latter part of the last century and it's the subject called statistical mechanics statistical mechanics is precisely a piece of theoretical physics that allows us to guess to know sometimes apparently to truly know in advance what these distributions will turn out to be there are wonderful results that have issued out of statistical mechanics coupled together with quantum mechanics some of our very best knowledge the nature of black body radiation various things that even led to the formulation of quantum mechanics it's an immense host of knowledge that works out wonderfully well in comparison to the measured behaviors in the world but if one looks and inspects what are the kinds of problems that statistical mechanics can solve then we discover that at the very part of this subject is a very important assumption the assumption goes under many names mathematical name is called eroticity has different sorts of names the answer is in fact one of utmost triviality the way in which the theory of statistical mechanics is put together is simply that if you look at all the different microscopic configurations all the different ways that this fluid particle can be here moving at that speed and do that for every one of them there are very very many such configurations that can correspond to a given macroscopic behavior in any of this size loosely sitting at that place well the rules of statistical mechanics is that each one of these my microscopic configurations is exactly as apt to occur as any other and that's fundamentally the entire backdrop of statistical mechanics if that statement proves to be true then you're in business it doesn't prove to be true then you're not in business and you have to do something else the problem is that if it isn't true our mathematical knowledge today does such that fundamentally we can't do anything at all so this is the basic problem we know what we want to do we want to figure out an appropriate distribution function say for a turbulent fluid or from others some other such problem that will allow us to produce our statistical average predictions which we could then confront with experimental measurements but we are unable at the moment to figure out how to do that as a matter of fact not only is it difficult to figure out how to do it is an extra problem and the extra problem is that we don't even mathematically understand what the nature of the distribution function that we're looking for is going to be if one would press at the moment you demand of me to say well what will it look like then I would have to offer a prognosis there will be something extraordinarily complicated it won't just be that it's something smooth with some bumps in it it will probably be something horrendously complicated a mathematical object whose nature we don't understand at the moment as a matter of fact whose nature we don't even know amongst what class of things it's to be sought amongst if it should turn out to be a fractal object and with some rather simple generosity of thought almost all complicated things of these shape like geometric structures could be termed fractal then almost certainly it will be a fractal object beyond the description that we now know how to perform and so at the present moment one is faced with a challenge that one doesn't really have the vaguest ideas to how to proceed in doing this problem let me just show you some pictures very quickly to give you an idea from vastly simpler problems one of the sorts of things we encounter which will give you some idea as to why my prognosis is perhaps sad here's an example is this working yes here's an example of a very simple object what you're looking at over here is the behavior of a very simple mechanical system this mechanical system consists of a cylinder just a hollow tube sitting inside the tube is a piston so just the disc that perfectly fills up the side of the tube it can move up and down and the piston is being supported by a gas so of course if you have a tube closed at the bottom with a piston on top and this air inside the piston will come somewhere to settle at a certain place it won't fall to the bottom because of course there's the presence of this gas which can exert a pressure to push the piston back up well let's drastically simplify that problem and replace the gas by just one molecule so this is a picture that has to do with a very easy sort of well known problem it's a problem of a piston being supported inside a cylinder held up by a gas but the gas consists of precisely one particle well a natural thing to do rather than to show you how this gas particle is moving because basically most of the time it's just sort of moving straight until it bangs into the piston and reverses its direction so what I'll do instead is only look at those moments when the gas particle hits the piston at that moment I'll put a dot in this picture according to how high up the particle was which is the same height as the piston and what its speed was and if I accumulate all of those points so each point here gas particle has gone down hit the bottom of the tube come back up and recollected with the piston each one of those points is one entire circuit of this motion and if you look at how these points lie you'll immediately observe that in no way do they fill up this whole plane in order for a statistical mechanical treatment to work it must be the case that this surface should have been uniformly perfectly filled with points so here's a very simple example where if one asks out of this problem what is the statistical behavior that you expect what is the average height if you want to the piston what are the fluctuations about its average height those are questions at the moment no one knows that answer this is already a representative example of a problem whose average motion is already sufficiently complicated that we don't have the means to understand that you can see that there's a big hole here there are holes here you can see there are little holes elsewhere all of the points that look dense they're just one starting point we slide off the piston and the gas particle in a certain way we watch it bounce after bounce after bounce and that fills in all of these points starting it perhaps at this point it will run around these three although it turns out there are four copies of them so after 12 collisions it will come right back to the same point and is an immense variety of different behaviors that we would observe and so to average over them is something that is absolutely forbidding and as I said no one knows at the moment how to really do even so trivial a problem was that as another example this is perhaps slightly more reminiscent of fluids this is an example of the so called strange attractor that appears in a modestly elementary equation I'm not going to bother saying what one's doing again one is getting points on this picture because we have a smooth system it's a mass on a spring more or less it's running around however we only watch what it's doing every one second and we put down a point according to where it is and what its speed is every one second that we take a picture of it and we do that we see a reasonably complicated object if I wanted to compute the statistical behavior of this object it would have been child's play it would have been the statistical mechanics that we've inherited had these points rather uniformly filled up this space here I'm now pressed to answer a much harder question what is this object just how does it fill up this space how am I going to do the correct averaging only over those points that this particular object only visits statistical mechanics has happily cheated because it simply guesses in advance that it would have visited everything if problems exist for which that is true and there are hosts of such problems some of them extraordinary problems then you're in business as I said before in this problem you're not and again to figure out the statistical properties of even so reasonably simple an object is beyond all present means so that's an example of what things look like these are the most elementary sort of systems you can imagine if I tell you I'm going to think about fluid problems I'm not talking about two things one playing with the next I'm perhaps talking with a million of them and in that case obviously these pictures will get more and more complicated and so if one is going to guess from the simpler pictures what happens one anticipates that something indeed very complicated will happen as a matter of fact we know that we know that because if we directly take the theory of statistical mechanics and assume that the fluid can take on with equal probability all of its possible microscopic configurations from that we can go ahead although it's arduous and deduce what the full statistical theory should be when we do that we get very definite predictions they're very definite and they happen to be completely wrong so one knows very well that the program of statistical mechanics all the ideas that we understand in advance how to produce these distribution functions simply don't work in some of these problems if they don't work in these problems we're really in a serious bad shape in some ways the study of chaos and I'll say much more about this has at least tackle these problems directly and in rather more modest circumstances which is the rub it would be nicer if we could do it in in fully developed fluid circumstances at least in more modest circumstances they now put us in a position to at least know what to expect for the complexities of the simpler of such objects and in general it least gives us some starting results so that's sort of a backdrop having said these things I introduced my talk by saying chaos was a part of a larger branch of study that larger branch of study being called the study of strongly nonlinear systems and so let me now backtrack and tell you what nonlinear means and what strongly nonlinear means in order to do that since the word is non something it's better to start off with the positive thing and let me quickly say what linearity means linearity in physical terms is a very simple statement what it means is that the rules that determine what a part of an object is going to do next are independent of exactly what that object is doing now that says no matter what the object is doing where it is how quickly it's moving if I want to figure out what that piece is going to do next I don't have to pay any attention to that information in a more precise way this idea of being independent of what it's doing is only intended in a differential or if you will incremental sense and so a very good example of that is so-called linear spring for a spring there are two obvious things to talk about one is the extension of the spring how far have I pulled it the other is how much is the spring beginning to pull back on me which is called the tension in the spring if the spring is linear that's simply the statement that for an increment of the length of the spring there will be an increment in the tension that the spring is pulling back on me in just such a way that the ratio of those two increments is always the same so no matter how far the spring was already extended if I increase it another one inch I will get exactly the same increase for the amount of tension the spring pulls back had instead I started with the spring on extended and also pulled at one inch that's in immense convenience of description obviously because it means I don't even have to keep track of where this thing is in order to figure out what the rules are that will determine its future it should be abundantly clear that a real spring can't be a linear spring the reasons very simple if it's true that the increase of tension is always the same no matter how far I've extended it it means amongst other things I can arbitrarily extend that obviously a real spring will snap at some point and from that it obviously follows that a real spring in fact can't be linear in general most of the things we encounter are not linear things well why is linear so important as I said qualitatively thinking physically there's clearly a simplicity that you don't have to understand what the thing is doing now in order to specify what the rule is that will dictate its future however it turns out there's another reason that one likes linear things and the other reason is deeply geometric in spirit if we think about the mathematics of linear objects it turns out that there's a truly elementary if you want very felicitous geometry in which one can understand all such problems as a matter of fact one of the larger bolts of the mathematics that we've inherited from the last centuries is the mathematics that teaches us and makes us arbitrarily powerful in coming to understand the solutions of these linear problems so the linear problems are nice because they have a very easy geometry even if sometimes it's a little bit difficult to get a number out nevertheless is an absolutely clear simple mental image that allows us immediately to know what's going to happen to figure out contingencies to come to figure out the nice things to think about for that object and in fact that can allow us to solve them with almost arbitrary ease that situation persists so long as there are finitely many parts the details get more complicated the picture only changes when they're an infinite number of parts the theory of quantum mechanics if you want is a linear theory with an infinite number of parts to it even in those circumstances one can find this geometry still so felicitous that one can easily produce exact answers so the important point here is that linear from the sense of mathematics means one universal very elementary geometry not linear amongst other things means the opposite to that doesn't always and I'll have much more to say about that but what it does mean is that first of all no very felicitous geometry exists even worse than that and this was the standard presumption many people certainly still feel that way it was the standard thought over a decade or so ago and that is that each nonlinear problem presented a new problem in its own right each of them was thought and sometimes does if you want to produce its own geometry these geometries differ so significantly from one to the next that it was felt loosely impossible to make general statements about them and many of the advances in dynamical systems many of them associated with the name of Smale have very much to do with understanding certain important ways in which these geometries are at least not anything you want finding very general cases of geometries where one can make rather strong statements the upshot of that for physics is that you might identify a nonlinear problem a certain flame propagation problem if that problem was sufficiently interesting to you there wasn't any mathematics that existed to solve it you could try to produce new tools that particularly would work on that problem however everyone who did that knew perfectly well that those tools would be of no interest in the solution of any other problem and so one had this natural prejudice in terms of thinking about nonlinear problems that whatever the effort you are going to put out was the upshot was that if you were lucky enough to be successful it anyway would be an insight a set of techniques that would work in that problem only and so here we have this problem we have these nonlinear objects they possess complicated geometries and we have to ask ourselves a question how are we going to come to understand what these geometries are if you want you can think of a certain kind of historic precursor to that problem is a problem associated with Gauss and Riemann namely you can try to think what is the geometry on the surface of the earth look like we're rather potent as human beings we naturally envisage three dimensions and so we usually envisage the surface of this sphere as something immersed in those three dimensions however if one wants to understand more generally what happens on curved two-dimensional things it behooves you to figure out how you can intrinsically describe it without reference to this bigger space how do you figure out what's happening just living on the surface of that sphere and that solution was first proposed by Gauss and then extended to higher dimensions by Riemann and so in some way one is facing something of a similar challenge one is seeing objects that are producing very complicated geometries such as some of those pictures that I showed you how is one going to start figuring out what this geometry looks like how does one quantify it perhaps when it's important how does one discuss it in an intrinsic fashion these are the questions that we certainly need to come to understand and I'm raising this as a question because indeed it is one of the successes of the study of chaos that one at least in limited cases has come to understand how one can implement such a program and indeed succeed in doing it well as I said one of the prejudices against nonlinearity was that each nonlinear problem was viewed as different from the next should it turn out and indeed it has turned out that very many of these different problems share the same geometry then of course we're in a position to start examining the general abstract problem the shared geometry might be only in some quantitative sense rather a qualitative sense typically there might be something that's topologically common in various of these problems it might also turn out however that the shared geometry turns out to be exactly the same geometry even when viewed in a quantitative way and basically what I'm going to tell you about is that set of circumstances that we now understand in which many of these nonlinear problems indeed exhibit the same geometry and the geometry is not only qualitatively the same but when expressed as an intrinsic geometry just on the motion that you're concerned with itself then moreover that geometry quantitatively is one in the same the upshot of that is that all of the statistical predictions in fact even much more precise predictions than statistical ones will be exactly the same for all of these different problems this idea that quantitatively the results will be identical is the thing that we term universality so universality simply means that forget about any details even though they're very different objects they have different equations nevertheless the solutions with respect to the right questions are one in the same identical in order to say something about that let me begin to offer a qualitative argument as to why you might come to expect this property of universality the argument that I'm going to offer is more or less a verbal rendition of what the mathematics looks like and I'll just try to give you some idea of what it is that's at stake if I look at some nonlinear problem what that means is that if I look at a part of it and I ask for a new increment in its motion what is the increment in the forces acting on it it will no longer be the case that that ratio this coefficient of proportionality is always the same what it means that it's nonlinear is that as a matter of fact it will vary it will vary according to how far this particle is distant from a starting position it will vary according to the speed of the particle it will vary according to what other particles are doing well if you think about that there's also something of a bonus in it because if it's true that depending where this thing is what it's doing it's effective at the moment coefficient of proportionality that will dictate its future depends upon where it is then we can think of actually setting these constants to be things that they that we would want them to be simply by presetting the motion so loosely an idea merges that if you have a rather complicated object which is nonlinear by appropriately setting its parts into motion you might be able to achieve something desirable but I have in mind more specifically is considered two completely different problems one of them perhaps is a laser physics problem another of them has to do with the interaction of a large number of different chemical species these are both of them nonlinear problems I might ask that by appropriately setting some constants in the first place for example if I think of the laser problem one thing is how much energy do I put in in the bursting power of the laser in the chemical situation what density should I guarantee exist a certain of the species there are certain constants in the description of an object that has more to do with the environment than the object itself these constants descriptive of the environment mathematically are things called parameters there's some number of them we can imagine beginning to adjust several of those to try to make the behavior of these two completely disparate objects close to the same had the problem been a linear problem every one of these proportionality constants between extension to displacement you could view all of those as parameters and so by adjusting all of them I could take this problem and make it exactly identical to that problem so if I have two linear problems with the same number of parts and same connections if I'm willing and allowing myself to adjust enough things I can make one of them exactly the same as another one of them of course if it entails adjusting a million different things that isn't especially interesting this idea that since the constants naturally can adjust themselves in a nonlinear problem contingent on what the thing is doing perhaps here opens up a possibility the possibility is that yes I will try to set a few of the constants but now by judicious arrangement how the different parts of to be deployed with what speeds different in this problem from the way I do it in this problem there's then a possibility that in fact these two could end up behaving the same way the only question in the course of doing that is again how many parts do I have to adjust if the parts are innumerable or at least very large that's it best or rather pedantic offering well I'm now going to add to this a second argument and the second argument makes this first qualitative argument quite powerful imagine that I'm thinking of a particular kind of motion for these different nonlinear problems and that particular kind of motion has come about through the imposition of a succession of qualitative constraints so what I'm going to imagine is that first I see one kind of motion I'm going to constrain it so that it does something rather regular and if I'm lucky it's something that this object doesn't mind doing I can now try to change things a bit and impose a second qualitative constraint again I need to assume that I shall be lucky that these objects are happy they are amenable in carrying out this request to have some number of qualitative requirements set upon them imagine now that I'm very lucky and in fact can impose an infinite number of these qualitative constraints or requirements upon the system well sometimes it turns out that that can happen this is the missing part of the argument it's if you want remarkable that that can happen nevertheless if you find yourself in the circumstance for which for these two disparate problems each of them can be forced to subscribe to an infinite number of qualitative constraints then in fact you're really in business because here you try out the monatology of Leibniz and Leibniz exists exactly the same question the question is if you have an object for Leibniz it means the universe if you want which is self consistently constrained by an infinite number of constraints how likely is it to be that this object could be self consistent and then following Leibniz we can hope for the answer that there is a uniquely one possible way in which this could happen well the upshot of the theory that exists to describe some of these behaviors is precisely that by capitalizing on the non-linearity to find certain prearrangements of completely different objects we can start them off in ball parks in which they're sort of doing the right thing then indeed it turns out by adjusting one entity in the environment of each of these systems one can force it to be subjected to more and more the same qualitative constraints by the time one has done things just right which turns out to be very very easy to do it will be infinitely constrained and the mathematics indeed dictates there is a unique quantitative way that that motion can then unfold so as a qualitative argument one has some reasons to believe that nonlinear systems and from what I've said in the first part of the argument really only nonlinear systems can be expected by rather easy adjustments if you're lucky to end up doing one quantitatively identical sort of behavior and that then is this idea of universality well gone way past wherever I was sitting over here let me say something which is also in the way of a backtracking I'm talking about different kinds of nonlinear problems there's something important that nonlinear problems come in different kinds if you want there is sort of very easy nonlinear problems and then there are much harder ones which are these ones that I'm calling strong nonlinear problems there's a little piece of easy intuition which is worth mentioning at this point if I look at one of my linear problems if I look at this linear spring increase it a little bit further in extension I'll see exactly the same increase in tension I can make that a nonlinear problem in a truly easy way all I need to do is put on a pair of glasses that distort things if I look at the same object through distorting glasses well first of all when I extended a certain distance because of the distortion the observed increase of extension is going to be different contingent on where in the visual field this object is as for the increments they're of course unchanged you measure them off a strain gauge or something so the upshot of that is that you can start off with a very easy linear problem and just by putting on distorting lenses you can turn it into a nonlinear problem these are the kind of problems that are clearly the easy nonlinear problems if I should think about the kinds of problems we can handle as physicists and I'll make more comments about that in a second then as a matter of fact it turns out we've really only been able to handle the ones that come about by distortion and so in high-energy physics in astrophysics in all the kinds of physics and other analytic disciplines we own there's a unique method that we know how to use that we've been trained in using which is a so-called perturbation method this is a method whereby starting with an easy approximate answer we can successively make corrections until we have an arbitrarily accurate answer if you ask when this perturbation method works the answer is very easy the perturbation method works only for those nonlinear problems that are distortions of linear ones well that's an important point the problems that are called strongly nonlinear problems are exactly the kinds of things which do not appear in distorting glasses and so if you see a phenomenon one of them happens to be chaos if you see some such phenomenon that is not what you would see by distorting a linear problem then the various methodologies that we've inherited simply failed to work one of the reasons that the study of chaos is particularly interesting to a physicist is that we don't know how to solve any fundamentally of the problems we truly care about we own some mathematical machinery which is completely inept and inappropriate for using in the kinds of problems we would like to deduce some consequences for the study of chaos is in this sense part of the study of nonlinear problems and it simply means we are desperately trying to understand what is the right mathematics what are the right techniques when we are presented with these problems it's very easy to give some ideas of this there exists presumably a very good theory of high-energy physics this is the so-called standard model it embraces what's called quantum chromodynamics it's a very definite theoretical framework no one knows how to compute from this theory what its predictions are other than for certain symmetry predictions one thing one would like to know for example knowing something about quarks is what is the mass of a proton an easier problem is what is the mass of a pie mason well it's worth noting that after a decade of immense effort by very many physicists using immensely powerful machines not only the supercomputers that exist but specially crafted more exotic ones no one can still give you an answer to that question that puts one in a rather funny position you can ask is the theory right but you don't know how to calculate the consequences or is the theory wrong at the present moment we have no way of understanding the answer to that is going to be a next conference here next year professor Mandelbrot has shown you some pictures of the distribution of of matter in the universe galactic distributions can ask yourself the question who knows how to calculate calculate not measure what this distribution should be who knows how to figure out what the evolution of the early universe should be if the distribution of matter isn't absolutely isotropic to date we have no tools fans of questions like that the second question is even difficult to figure out how you can stick on a computer because of some subtle difficulties of the underlying theory so one is always faced with problems where one needs machinery for which we simply don't have any ideas as to what they are and so if perhaps the work that we've done in chaos the onset of of chaotic behavior the transition to chaotic behavior is perhaps simpler than the questions being asked its point and its aim is indeed to start furnishing some of these missing pieces of machinery that we desperately need well see if I'm not showing you any pictures I haven't said anything so far about what chaos is I haven't said anything about the transition to chaos at least I've told you what nonlinearity more or less is about one of the things that one of course must have heard about if one has heard anything about chaos is that chaos has something to do with sensitive dependence upon initial conditions as I've alluded to made hopefully a strong case for not only is one interested in knowing the answers to questions like that but then is a general enterprise in the study of nonlinear problems whatever the outcome of the answer to a question of that sort is one cares and inventing new mathematical machinery nevertheless one does talk about sensitive dependence on initial conditions as having been the main thing that one talks about in chaos technically when we say something is chaotic we mean by that that we're looking at a system which has the property that errors in it grow very rapidly technically we expect them to grow exponentially quickly this motion of sensitive dependence on initial conditions is exactly that statement and so the statement of what it means for an object to be chaotic technically is the statement that there is a very strong growth of errors of course the strong growth of errors inhibits the ability to make predictions even though the underlying theory may be absolutely deterministic well that's what chaos is I haven't told you what the transition to chaos is well you can figure out from the words transition to chaos that it means somehow that first of all the thing isn't chaotic and then it's some later point it is chaotic and somewhere between those two something has happened that's produced a transition from the non-chaotic to the chaotic behavior the remarkable fact of what we have learned about some of the ways in which things become chaotic is that they become chaotic if you want at a precise well delineated moment there is a very specific kind of behavior a precise crossing over point with something moves from not chaotic to chaotic at that crossover point which is this transition to chaos the system is neither non-chaotic nor is it chaotic what that means is yes errors do grow so it isn't non-chaotic however they don't grow as fast as you would expect with the idea of sensitive dependence upon initial conditions technically for those who know the distinction there is an algebraic growth of errors at a transition point whereas there is an exponential growth of errors once one is seeing genuine chaos so the first point is that we have this circumstance in which is a transition non-chaotic chaotic at the transition point what we're looking at isn't truly chaotic itself well what is it that happens at this transition point well it isn't chaotic however the most important part is that sitting in this behavior this one precise behavior at the transition to chaos appearing if you will in embryonic seeds is every possible behavior that these systems can exhibit the way in which this transition occurs even though it is not chaotic is such as to construct things if you want with the right kind of inheritance heritage so that just past that point the progeny of that point will know precisely how they are to behave truly with chaotic or sensitive dependence on initial conditions so as a matter of fact the property of sensitive dependence on initial conditions is already inherent in this embryonic fashion at the transition point mathematically the situation is perhaps stronger than that for some of these problems what happens that transition point essentially completely organizes everything that you can see in the problem and I'll return toward that at the very end well next thing is what is it that turns out to be universal what turns out to be universal is a certain specification of the intrinsic geometry on which these motions occur and at this point it's probably a good time to show some pictures and try to I hope this works does if I lean to it we are seeing here are some rather simple depictions of a simple one-dimensional problem which is intending to show what these seeds look like what you're seeing over here this is a very simple problem it's time is a discrete time you're watching something move what happens one second later is dependent on what happens now it moves only in one direction all of where it can be is between these two points and what you see collecting points to make a histogram of how often the point finds itself in what place we see some sort of a distribution this is one of these distribution functions that I've been talking about if we adjust one thing in the environment of this problem which is one parameter then at some point instead of being one entity this distribution breaks up into two things in fact I've showed you here a transitional moment it's getting very very apt to be at this point if I change the parameter a little bit suddenly it breaks apart and leaves open a gap at some point the gap is now very big either it lives in this part or it lives in that part this value of the parameter I've chosen is such that again each of these is in turn also getting ready to break apart you might notice that if you take this thing and you flip it around and squeeze it in you'll get this picture if you take this thing and you just squeeze it in a lot you get that picture well this was chosen so that if I change the parameter a little bit more each of these will break apart well the next picture here shows this one broken into two pieces just as that one did and similarly this one has broken into two pieces well you can do this again and again and again worth noting that this piece again is almost identical to that but it's been squeezed down still further again it's this whole picture flipped over reduced in scale that produces this picture the process that I'm looking at is a process which is natural to many of you which is the process of constructing a counter set the idea is we start off with one thing and we make a deletion here I've deleted out this piece so now I have two pieces left over at the next step I again make a deletion a big one here a skinny one here I have four pieces now I can make a deletion each one of those doubling the number of pieces of those which produces this last picture I'm showing if I do this again and again and again in infinite number of times when I'm all done I don't have any actual fat intervals left I have what's called a counter set but that counter set is exactly those embryonic seeds I was talking about it knows if you will precisely what every one of these following pictures must be because it's transitional instead of having sensitive dependence on initial conditions these are truly all chaotic behaviors it has this more marginal crossing over a sort to say more precisely what this geometry is that becomes universal the geometry of course is to know intrinsically how do you make up this counter set a way of providing that information is everywhere along it figuring out how big a hole should you burn in where should you put the hole and how to do that no matter where you're sitting by the time you've done it a lot of times and there are many parts it would have been very easy if the rule were always the same however in these problems these dynamical problems it always turns out that the rule is different at each point that you look at and so if one thinks of things such as the simpler objects that professor Mandelbrot showed you some of the snowflakes gaskets carpets and whatnot those are all made of one or two or just several different ways in which you change the size of decorations as you go down in scale all the dynamical problems that I'm aware of that arise in these more complicated contexts always have the property that there are an infinite number of such scales so as one comes to learn the geometry of these objects better one is learning more how to describe more and more complicated fractals to use that word me show you a little more graphically what one of these transitions can look like this is drawn from the same rather simple problem the problem for which I showed you that funny curved around so-called strange attractor here at some appropriate value of a parameter you're seeing a motion that runs around on this every one time every one second comes back to exactly the same point and you see just one loop in the picture this time I'm not showing you only one second which would be the blue dot I'm showing you the entire motion well if I change the parameter some at some point that one loop becomes unstable system doesn't want to do it anymore if there's any perturbation it stops doing it and instead now it breaks apart into these two curves that is one time around starting at this point it pursues a path and it comes back to this point it takes an entire new circuit until you come back to the starting point so what's happened here is that we've taken one loop repasted by a pair of loops this is existing in a three dimensional space there's a twist that's been put into the picture the geometry is no longer quite so trivial this clearly also a rather uneven spacing as you look at this picture moving along the two curves well if I change one of these parameters still a little bit further then I'll see a picture that looks like this there are now four of them the way you should think of this picture is not that all of a sudden we got four what's happened is that originally we had two each of these single bands in the two case has now in turn split into two and that produces if you want two approximate copies of the same thing this picture already tells you that this something very nice in the geometry because what the picture shows you is that when on the average these two pairs a far apart the far apart here becoming narrower over here if you now look at the internal spacing you'll see that similarly it starts off wide where this one is wide getting narrower with the overall separation is narrower and so you can ask yourself the question is it true that there's some definite way in which how this more complicated motion can be viewed as a decoration of the previous motion has a simple way in which it's a scaled down copy of the previous motion if you imagine cutting this by a lion here you're again seeing one of these canter like things originally we had one distance now we have two distances because we've created a hole in the middle and as you might imagine this phenomena will continue again and again and again well I can try to actually do that run it off on a computer I could do it for a physical problem I could look at perhaps a helium cell the world kinds of things I can look at many things like to do this behavior the qualitative constraints that I was talking about is I started off by seeing this thing make one execution of a loop by adjusting just one number I made it to do two executions of a loop by another number for executions of a loop each of these imposes a new qualitative constraint on what that motion is and the assertion that I was making is that if indeed it can then subscribe to an infinite number of these qualitative constraints then I have every reason to believe that the outcome will be quantitatively universal so what I'm going to go and do is to measure the broad separation then measure the finer separation and everywhere around this by the time it's a lot of times ask what the ratio of those two is well if I do that I see a picture that looks like this the lower half is not of any particular importance it's just the top half made negative it looks sort of messy but you can see sort of there's a piece here a piece here a piece here it sort of looks like some more or less straight parts with some jumps in it in fact it's rather surprising there is so few jumps I'm looking at this distance running around on this band it's rotating around from time to time the distance is turning through zero if I look at any projection of it there are lots and lots of zero crossings the fact that there aren't more divergences in the picture when the denominator is equal to zero already indicates that something rather nice is happening well as I said this is leading to a certain universal behavior is a full theory for that and the answer is it must look like this this object tells you everywhere as you run along on this more and more divided object as we're producing more and more bands what is the size of the hole that I burn into the piece how much smaller is it than the starting part and how does that vary as in time I run along this whole trajectory well the theory says it should look like that and you can sort of see already that there's a rhythm legal agreement between these two and in fact you can look at any sort of problem you want that qualitatively does this successive doubling one band two bands four bands if you ask these right questions what is the nature of the geometry on which it's living specified by this so-called scaling function which tells you directly how you refine pieces in the limit of infinite refinement you will always end up seeing precisely the same geometry the upshot of that is that the motion in the right way is universal that is the numerical values of universal if I look at the power spectrum things that produce for example one over F noises that professor Mandelbrot was mentioning here they're not one over F noises if I look at these they too are universal as another example a different geometry here's what one of these power spectral look like this is for a different problem it's a problem of quasi periodic motion power spectrum means for those of you who don't know it you might know that a musical tone consists of a fundamental tone and several overtones or harmonics it's a wonderful fact of mathematical analysis that anything whatsoever that repeats itself after some period of time can be written as a sum of these fundamental different pure tones and what this picture is is simply telling you which pure tones appear in synthesizing this motion and how strong each of those tones is and it's a rather elaborate picture you can see that apart from going down this way sort of looks the same in bands each one turns out to be the golden mean times the next and size for this particular motion it's plotted logarithmically so constant multiples mean constant sizes and this clearly some regularity to the picture is again a complete theory for this motion which produces another one of these scaling functions by brutal approximation just to make a calculation easy you can very easy calculate what its consequences should be produces this object this is now essentially a trivial exercise you can do this almost on an adding machine you barely need a pocket calculator you can see that while this isn't exactly the same thing qualitatively we're obviously talking about the same thing two of these something dipping down in between one wants to do a better job one could do a much better job well that can be contrasted finally to some real physical data this is data of Albert Liebschaber this is a certain experiment of mercury which is being heated and which has a magnetic field present in it for perhaps inappropriate reasons the way this data was presented was that the way everything drooped there was factored out so now everything has the same size we should be able to see that there are bands present this band this band aren't too many of them they were drooping down things inside them and rather evidently one is looking at the same sort of object with better data with some more details put in just from a theory that universally says the geometry must be this one can reproduce all of these experimental data so rather striking comment one of the virtues of this theory of universality is that while we have good candidate equations such as Navier's folks equations doesn't mean they're the right equations in fact we know they're the wrong equations we don't even know the right ways truly to put in heat flow there are lots of different candidates and approximations this universal theory says it doesn't even make a difference you don't even need to know what the right equations are in order to determine what the solution should look like if you want that's one of the striking qualitative say philosophical consequences of this sort of work what I mean by that is that historically we as physicists have been taught that qualitative thinking is nothing that one can ever pin truth upon quite to the contrary what we're taught is that we must go out and experiment we must have amongst our different candidate theories have worked out exact and precise consequences go to the world inquire which is the right answer and then keep doing that until finally all but one candidate has been eliminated and that candidate should better perform extraordinarily well or else we say we don't have the right theory here we have a rather different result because here it turns out should these things be qualitatively the same will be it with an infinite number of qualitative constraints then in fact we already know the answer the quantitative aspect automatically here issues out of the qualitative one so there is something that's clearly somewhat remarkable about this it's worth mentioning that when the theory that explained for this problem of doubling what the nature of this transition was theory appeared in 1976 I was instructed by some colleagues to go and tell these things to a rather well-known mathematician who is rather famous for some of his work in dynamical systems this person is not present today so he won't be embarrassed and I try to explain to him that one was now in a position to assert that there was a quantitative universality in this phenomenon of doubling well each time I tried to tell him that he fired back to me but we already know that because of course someone or others qualitative theory was what he had in mind finally after about three interchanges as he grew more frustrated he said you mean to tell me these are metrical results metrical by the way is mathematical code word meaning quantitative and I said yes and he said well then you're wrong and he walked away so these were completely unsuspected properties the qualitative properties remarkable in themselves at least had been developing for a long a period of time this quantitative universality was a rather new object as I mentioned and I'm essentially done this object that determines the geometry the intrinsic geometry in the doubling problem it loosely has just two values to it you look a little bit harder it better than that has four values each of those in turn are different so it better still has eight values as a matter of fact there are an infinite number of such values a rather simple sort of fractal could be represented by having say just one constant one of these values perhaps two constant ones and so forth however while this has an infinite number it's at least reasonably close to some rather simpler ones there are many problems that look nothing like that so to show off a few more little examples this is a so-called devil's staircase what it represents is that there's a knob you're adjusting and as you're adjusting the knob things come into synchronism or they don't and so as you're adjusting the knob this way there's some range over which you'll see the things synchronized the model you should have in mind here is a phonograph turntable asynchronous motor there's a range of torque on the motor for which it will continue to move exactly at 33 and a third revolutions per minute but if you put on too much torque on it it will finally stop and perhaps do something different so here is we adjust some knob increasing the torque there's a range in which we see some rate of revolution then suddenly it jumps to something else and we see this staircase we can ask about what its geometry looks like if we look at this intrinsic geometry we see a picture that looks something like this and now it's rather clear that what isn't seeing in any decent sense something like say two constant values is truly here a very large spectrum of different ways that these different steps are produced one fractured from the next you remember what that picture looks like here's a different problem closely related to something that Mandelbrot had showed you this is not the Mandelbrot set it's a simpler object it's a certain so-called Julia set and sort of looks like the outline of a cloud it's a rather easy to come by object it occurs at one point that appears in the Mandelbrot set it looks something or other you can see there's a piece in here that sort of looks like the whole thing if you take this piece and flip it over so that part moves over here then there's this bump over here and it pretty clearly looks sort of as though itself similar so here's a magnification of one of those pieces however to put it together isn't quite the same however if I now take this piece and magnify it up again and keep doing that it will finally become something definite the rule of doing that magnification however is different everywhere along this curve if I figure out again everywhere along the curve what is that magnification which is how to specify intrinsically what its geometry is then I get a picture that looks like this this is another one of these scaling functions determining the intrinsic geometry and you can see it's highly reminiscent of the object that you saw for that mode locking problem that devil staircase picture so in fact there's quite a large diversity of these geometries to understand these geometries is at least to completely understand some of the dynamical behavior of perhaps easier of the problems that we know about well I think I very close to to finish what I want to talk about the basic thing that I've said is that one has come to understand in the course of this study of the onset of of chaos the transition to chaos the nature of the geometries of certain objects we understand how we figure out that they universal there are strategies that we've learned a very important thing that we've learned is that indeed they are universal that has the crucial import to it that rather than solving a hard problem such as the fluid problem which is even exacerbated by not even knowing the right equations we can with impunity for the right kind of qualitative behaviors substitute a trivial problem once we've done that we can almost certainly make greater inroads into figure out what's going on in fact this idea of substituting simpler problems is one of the strategies that we've learned in the course of pursuing these ideas as a mathematical point while for the simpler problems there might exist more mathematics again there are so-called topologic properties that one understands very well of these simplest one-dimensional problems those topologic features turn out to have a much richer existence than anyone knows how to prove at the moment we only understand why that so because we know that these universal behaviors can occur in immensely high-dimensional settings hmm well what I'm saying them is that we've learned some method of describing some geometries it's more than saying geometries in a certain way of speaking when I say geometries I mean very specifically I can tell you how all the pieces are put together I can specify one of these objects by telling you exactly what its scaling function is these are entities that are in virtually one-to-one correspondence in fact are well let me end with an example you've seen pictures of the Mandelbrot set you saw them in the previous lecture many I'm sure have yourselves gone off and simulated the Mandelbrot set on your computer however I doubt that many of you are aware of the fact that the ubiquity of the Mandelbrot set is in fact the consequence of the idea of universality at the transition to chaos more than that as I said there were potent seeds that appear at the transition to chaos and in those seeds as I said is everything that can possibly happen so as a matter of fact the entire Mandelbrot set is organized by just that one point which is at the end of this hierarchy of doublings the most obvious way that you can see something about that is that of course the set is big disk the cordyoid one with another disk and a smaller disk and a smaller disk if you look along that part of the Mandelbrot set and you ask how quickly did these discs diminish in size then the answer to that is in fact they diminished by a number called Delta which is around four and two-thirds which is one of the most prominent numbers known from the study of the universal transition to chaos in the so-called circumstance of period doubling it's remarkable albeit true that just understanding exactly what happens at this transition to chaos in principle determines all of the things that you can happen nearby I'll just put up a final picture if you look at the seeds in the theory theory is reasonably complicated it specifies some appropriate function this function exists in the complex plane has a real and imaginary part however that runs if you don't know what it means it doesn't matter this is the real part of that function in the complex plane function is immensely complicated it has essential singularities natural boundaries floating around in it in its full structure in fact is everything encoded not easy to get out of it but nevertheless encoded what the entire arrangement of the Mandelbrot said is and so let me end my talk with the final comment maybe what we understand is somewhat on the simple side it is I would obviously like to be able to tell you this is the way the fluid turbulence works there are a lot of things I like to be able to tell you there are approximate applications to biology when those many things now about breathing diseases apneas various a rhythmic problems one would like to know much much much more about them one has very limited knowledge at the moment nevertheless that simple knowledge perhaps pushes you to think that along these lines with more effort more distortions one might be able to comprehend vastly more complicated things and certainly it's in the attempt to comprehend those much more complicated things that we find that justification truly in doing this study thank you very much