Panel discussion: The impact of chaos on science and society

Feigenbaum (chairing): In this session, we are going to try to identify those circumstances in which the research in chaos that we are aware of, has had an impact upon the physical and social sciences and then consider what the future might bring along these lines.

I would like to start off by reminiscing a little bit. In terms of the general subject of the conference, "the impact of chaos on science and society," there are a variety of things that induced me to start thinking about what are now called "chaotic problems." One thing that made a very strong impact on me was that approximately in 1972 or 1973, an author named Immanuel Velikovsky, whom some of you might recall, speculated that our planetary system did not always look the way we see it now and put forth a strange theory that Venus, in historic times, was in a different place. His writings, from time to time, have captured people's enthusiasm and in the early 1970s, following an episode toward the end of the 1960s, large sectors of the population became enthused with Velikovsky again.

At that time, since anti-establishment thoughts were generally rife, there was renewed interest in astrology and so various members of the scientific community, and especially the physicists, decided that the matter was getting out of hand. It was felt that there was a need to remonstrate to the populace at large that they were wasting their time reading nonsense. I recall, at some point in 1973, a big full-page spread in the New York Times signed by many well-known physicists. I think it was written by Carl Sagan, but I remember that Hans Bethe and Victor Weisskopf had signed it, as well as a whole roster of well-known American scientists. The points made were that, since we know that all questions of astrology are nonsense, you would better spend your money and time on more genuine science, and that this growing interest in Velikovsky was particularly foolish. Thus, these luminaries concluded that we know that the solar system is absolutely stable and we have scientific knowledge that refutes the claims of Velikovsky, so you are wasting your time.

For various reasons, some of them social, I did not think very highly of what my colleagues had decided to put into the newspaper. But I also wondered to myself, "How can it be that one knows that the solar system is stable?" So I started asking around and tried to do whatever calculations I thought one could do, when it struck me that, other than as a result of undue confidence based on ignorance, there was nothing that one could have known about that. As you have heard today, the situation is in fact quite different.

I think that it was Jack Wisdom who said that certain ideas - those ideas that we call "chaotic" now - have indeed changed some of the ways that things are perceived. And it is certainly true now that, if you asked Carl Sagan, he would not so handily say that he knows that the universe is stable and will be there for all time - the big bang discounted. With those opening comments, we would like to hear something about how experts in their own disciplines have seen chaos change some of the ways that research is done, about the results that have developed from considerations of chaos, and then, within the limits of what they think is believable, about the direction in which things might go in the future.

Peixoto: My background is in mathematics, but I am also a civil engineer. I had something to do with dynamical systems in my early work and, in a way, I saw those elements which started the application of chaos to what is called structural stability. There was a general feeling at the time that structural stability was very mature and it took quite a long time to see the beginning of an understanding of the chaotic behaviour of systems. Basically, the contribution that the study of the structural stability of dynamical systems brought out was the need to consider, not a single equation, but many of them as possible solutions, as well as all possible initial values for these equations. This is precisely the picture that happens to be so important and it is essentially a way of looking at the differential equations. And it was, in a sense, a surprise to me to find here that quite a few people have been interested in this approach.

From the talk of Steve Bishop, it is quite clear that this spirit is present at least in some groups of engineers. When he showed the picture of a ship with problems of stability and whether it was going to capsize, it was quite obvious that the boat cannot be studied as the movement of a point in space. He studied a number of neighbouring solutions and also perturbed the parameters. So that is exactly the spirit which dominated dynamical systems and which came of age with the work of Stephen Smale. This theory of dynamical systems is the very basis of this way of thinking and looking at the differential equations. So I would guess that chaos came about basically with the recognition by Edward Lorenz that weather could not be predicted through long-range forecasting. And it is not by chance that chaos was recognized in meteorology, because again the weather does not necessarily have a single trajectory to follow.

Also I would like to point out that parametric sensitivity, double periodicity, and so on are related to the problem. It is quite obvious that the solar system is definitely not chaotic if you look at it say over 10 years, although there are some exceptions with perhaps Pluto's orbit being chaotic even on that time-scale. But if you look over longer times, the solar system is chaotic. Thus, it is a matter of emphasis and you have to understand exactly how to choose the parameters so as to identify the chaotic behaviour. Looking toward the future, I would guess that we will see a continuation of work on chaos, given that there are now many journals on the subject. Of course, since science is subject to social phenomena, the study of chaos in science will follow the usual course of any new social phenomenon.

Ruelle: I think we must be thankful that most participants have been quite sober and honest in their presentations. Sometimes, we hear claims that the fractals of chaos theory will lead to a cure for AIDS. Such claims are not only embarrassing but have a bad effect on the image of chaos, and on science in general.

When evaluating chaos in a historical perspective, I think it is important to subdivide the subject. In physics and mathematics, it seems to me that the great period of excitement is probably over already. Chaos has been responsible for important results in the field of mathematics. In physics, one could make a similar remark. For instance, the theory of turbulence will never be the same now that we know that turbulence is chaotic, even though such a realization does not solve the problem of turbulence. Similar statements could be made about a number of other topics in physics. It is also possible that there will be a revival as a result of new ideas involving chaos in physics and mathematics, but I think most of the excitement is already behind us. Now, as far as astronomy is concerned, we have heard today that this field is now in the middle of the exciting period. As for biology, I think the exciting period is still to come. Why? That there will be an exciting period of application of the concepts of dynamical systems to biology is rather clear because there are so many quasiperiodic, or oscillatory, phenomena in biology. It is obvious that looking at them carefully from the point of view of dynamical systems will be useful and will lead to important results. It is just that the problem is very difficult from a methodological point of view and, therefore, to obtain important results we have to be patient. In this respect, I think claims have been made, not in Japan but in other countries, about chaos in biology that are unwarranted and dangerous.

Let me summarize. What is the overall opinion that one can have about the effect of chaos? I want to make a distinction between two types of result. There are hard quantitative scientific results that we have seen in physics, for instance in meteorology, those that are being obtained in astronomy, and those that no doubt will be obtained in biology. These are important. It is surprising to me how the idea of chaos has been so productive.

And there has also been an impact on intellectual activity in the sense of contributing philosophical ideas, which I think should not be overlooked. Whether we like it or not, such impacts are occurring. The word "chaos" is used in a non-technical manner by many people. But the fact that its use is non-technical does not mean that non-scientists have a totally wrong idea about what is going on. However, we as scientists can help people to have a better idea about what chaos is and thereby bring something new to their intellectual life.

Bishop: I agree with most of what David Ruelle says, of course. But also, I should put in a word of caution about giving quotes. We heard in this conference a quote of James Clerk Maxwell that is supposedly about chaos itself, but in fact was slightly misconstrued. I think the popular press often tries to find a quote with the word "chaos" in it, from perhaps 200 years ago, and then makes claims about chaos that probably are not true. I do not think we should bite the hand that feeds us, because the fact that chaos is popular means that more people are aware of the work we do. It is quite nice to have that recognition. But we must be aware that the use of quotes can sometimes be misleading.

Keilis-Borok: I am from a religious country in the sense that many people still believe that life has meaning. Traditionally, this makes people from my society very prone to distribute advice. So I will resist this and not voice any moral judgements about what is going on in the field of chaos, but will rather share with you my own experiences.

There certainly are advantages and disadvantages to getting on the bandwagon, because there is some transfixation due not only to the terminology but also to the pictures of chaos on the computer screen. These pictures are really hypnotizing and have the impact of looking at a fire in the woods. And you can get grants to look at these pictures. At the same time, chaos is a very revolutionary concept, and I see no harm even in the undue use of this terminology. You never can tell what a grant given now will produce later, and chaos may even prove useful to play around with. Thus, this concept inevitably penetrates one field after another.

It penetrated my field of seismology about five years ago and did so in a very typical way: there was some resistance because it is a synthetic approach to what people like to study at the level of minute detail. I think a very important warning is that the instruments of chaos theory - and nonlinear dynamics is only part of the picture - are not universal medicine, and should be applied only whenever it is felt appropriate. But in my field, it brought enormous deliverance because it raised research out of the literally bottomless pit of details to the level of rather simple and comprehensive pictures. But the only way to speed up this process is through education, for education teaches us to try out different directions, at least for a while.

However, it is important to keep in mind that the final judgement is whether eventually you achieve results or not. And these results should provide not just retrospective explanation but also prediction. While eventually all applications must pass this test, some applications are more urgent than others. I think it was Jerome Wiesner who said that the threat from the combination of natural disasters and technology is greater than anything ever posed by Adolf Hitler, Joseph Stalin, or nuclear war. Governments throughout the world have spent about $1 trillion per year to prevent this danger, most of which was for obsolete technologists, of course.

We are now in a situation where, in some fields, we require small Manhattan-project type efforts, not so much in terms of scale, but in terms of concentration of effort and intensity of coordination. Some of those problems are best understood by applying nonlinear dynamics. I think the problems of psychology, education, and certainly disarmament are urgent and, as we heard with regard to disarmament and psychology, nonlinear dynamics is relevant. So I hope that part of our effort will be directed to tackling these problems quite quickly. But I also understand that speeding up the application of nonlinear dynamics can occur only if the leaders and the team are viable.

Grebogi: I think chaos is ultimately mathematics. The theory of dynamical systems as such is being applied in the sciences, as it is being applied in technology. The initial impact was in the more exact sciences and in technology, and we saw that chaos in those areas was better quantified and perhaps better understood. But we also perceive that chaos is permeating the less exact sciences such as the social sciences and economics, although perhaps with results that are not as well quantified. As was just mentioned by Steve Bishop, you can even hear laymen use the word "chaos." Until recently, at a party when people would ask me what I do, I would say: "Applied mathematics." And their reaction would be: "I hate math"; and that was the end of the conversation. Now, when I give the same answer, people say: "Do you know about chaos? Can you tell me something about it?" and the conversation goes on.

On another point, I think this symposium truly attests to the fact that chaos is interdisciplinary and multidisciplinary, with mathematics at the helm. Regarding David Ruelle's question concerning what is influencing what, my feeling is that there is cross-fertilization between the sciences and engineering, on one hand, and mathematics, on the other, and that both sides are benefiting from the interaction. As Jack Wisdom has mentioned, chaos is crucial to understanding many lesser known phenomena in the sciences and engineering.

David Ruelle mentioned that there are not many applications. There is, however, something of an indirect application related to the launching by the United States of the space telescope in April last year. One of the problems is that, because of the Challenger accident, space shuttles are now heavier and cannot go up to such high altitudes because of the extra safety equipment they now carry. A competing problem in launching the space telescope relates to the current solar sunspot cycle. During intense sunspot activity, at the peak of the 11-year cycle, the upper atmosphere is heated and expands to higher altitudes. Because of the increased friction caused by a denser upper atmosphere, the life of the space telescope before falling back to earth can be reduced to only a few years if the orbit is not high enough. Many people worked to predict the intensity of sunspot activity, and the best prediction was made using data accumulated from 1700 and the chaotic dynamics technique of David Ruelle. It was predicted that the current cycle of sunspot activity would be the strongest in a century. And when the peak of the cycle was finally reached in July of last year, indeed, it was the strongest of all. NASA placed the space telescope as high as possible, to avoid having to send another shuttle back in a few years to boost it into a higher orbit at a cost of $200 million.

Feigenbaum: I think there is something that one can say in comment to some of these observations. Certainly we have seen the mathematics pertaining to chaos. Mauricio Peixoto made some rather strong comments about the notion of structural stability, and, indeed, structural stability has been one of the hallmarks of progress in this subject. But we certainly also realized that structural stability has its shortcomings. Certainly, from a numerical viewpoint, there is no way to really distinguish between a very long periodic orbit and one that is not periodic. In terms of structural stability, these are absolutely different entities.

Fortunately, various questions and methodologies can be used to treat the blurred distinction between these, which mathematically is not very easy to deal with. So one would like to see these in a somewhat enlarged scope. Again, for the general question of what is the physical input and what is the mathematical input, this has to do with societal traditions and the scheme of doing mathematics that has developed over the last 50 or 70 years, which has very much to do with extracting details.

But sometimes in the course of this general stripping away of unimportant details, one can strip away too much. So, for example, the notions of period doubling in the renormalization group treatments were alien notions to the mathematics partly because they truly rely on something relating to smoothness. But if one has stripped those considerations away, then such a phenomenon would never even appear as a general characteristic.

So certainly there has been an interplay of ideas, and very strong ideas relating to structural stability providing a framework in which to think about general problems of dynamics have been compellingly important in developing this field. But at the same time, as one sees more of what the spectrum of possibilities is, one comes to substitute certain nuances for others, and sometimes those nuances happen to be so dressed in the particular clothing of a given scientific problem that it would not have occurred to a mathematician to think about them because they are seen to be details. In this subject, I think there has been a rather honest interplay of purely mathematical notions in a style of doing things that is particularly useful compared to others.

Ruelle: I would like to make a comment on some prospects that look exciting. I think what we have gained with the notion of chaos is that we now have at least a possibility of access to some problems that are philosophically important and that until recently had appeared to be completely beyond our reach, such as: how contingent is historical development on small details that cannot be controlled, or is history determined in such a manner that it is predictable? Now this is a harder philosophical problem and there are philosophical views that support both possibilities. This is a problem that we would like to understand primarily for purely intellectual reasons, but perhaps also for practical reasons. Until recently, it appeared completely beyond approach.

Now, I think it is still inaccessible, but it does not appear as inaccessible as it used to. Our practical interest stems, for example, from the economic effects that can result from the unpredictable influence of weather beyond a few weeks, such as crop loss due to hailstorms. There is a further question about whether the effect of hail on a crop will have an effect on the economy. Although we are not ready to answer this question, at least we may now ask the question and start playing around with it. So, philosophically, these are all important but practical problems that are nearing the point of accessibility.

Feigenbaum: One of the questions that has been raised asks: is chaos a sort of flash in the pan of the same type as catastrophe theory? It is also important to ask whether our understanding of broadly related subjects, even of subjects now considered to be of a more philosophical nature, has improved and if we are in any better position now than we were when rather grand claims were being made for catastrophe theory? And when contemplating the claims made for catastrophe theory, we should perhaps remember the rather dismal history of a subject called "systems theory," which amounted to just using a lot of matrices. One of the wonderful results produced by this subject was that you should not build new housing because it ends up making less housing available. Such grand impacts seriously dominated the field and textbooks on economics in the early 1970s, if I remember rightly. So, we might try to determine whether we feel that we are better prepared and have better tools available to understand more deeply some of these very general questions, such as those Professor Ruelle has been alluding to.

Meese: I would like to pose the question in a subtly provocative fashion with the hope of at least being attacked by the mathematicians present. We seem to be moving to a stage where we can model systems by taking data, fitting nonlinear models to them, and then, with this non-explicit model, applying some of the results of nonlinear systems theory. There is nothing terribly provocative about this, but one could perhaps regard it as the end of the revolution started by Henri Poincaré. He essentially pointed out that we should give up the hope of obtaining closed form solutions to our mathematical problems concerning dynamical systems and simply give our answers in terms of qualitative properties. To go in this direction is to give up any hope of obtaining closed form models. Now, this is quite unnecessary in physics, at least at the level of the solar system, but if we are hoping to apply chaos to, say biology and perhaps the social sciences, then this would have to be the direction in which we should go.

Boldrin: I would like to make two comments. I think that I tend to agree, at least basically, with what I understand Celso Grebogi to mean when he says that chaos is mathematics. From my point of view, chaos is an instrument - a pretty good one and a very stimulating one because it provides you with alternative patterns that you may not have imagined before. But it is fundamentally an instrument and as such I tend to think that it need not be transformed into a revelation, religion, new age of science, or anything of that sort. It is quite powerful but it is not theory-free. It is not by merely introducing the techniques of nonlinear dynamics and chaos into a scientific field that we get marvellous results and turn everything upside down.

Chaos is an instrument and, when you introduce it into your field of research, either it has to make sense within the paradigm, or, if you are lucky enough, it provides you with an alternative paradigm. This is often the case in the social sciences, which I know a bit better than other fields.

Unfortunately, social scientists, especially those who do not have a tradition of formalizing their activities, have a strong tendency to grab anything coming from mathematicians and physicists that sounds fascinating, alluring, and has nice pictures, and they transform it into a theory of social organization. I think that this practice is quite risky. I do not think it provides much improvement in our understanding of the way that social organisms function. I would insist, in such cases, on the necessity of working within the paradigm to see if chaos theory is a useful instrument for solving some of the problems.

I am not sure that I agree much with the theory-free approach because I do not think it provides answers. It is true that it may satisfy Vladimir Keilis-Borok's requisites that a science is good if it predicts. But then, from certain points of view, it turns out that if you take that approach to the very end, particularly in the case of social systems, it reduces science to sophisticated statistics. I am not quite sure that making a good prediction by itself is either a necessary or a sufficient condition for a model to be good. Certainly this is not the case in economics where the problem of understanding what is going on is still more fundamental. After all, if you are working in financial markets, there are sophisticated statistical techniques that do provide you with lots of forecasting for the range you care about. Nevertheless, these explain little about what is going on. So I am not sure that I would be happy to see the role of chaos in the social sciences transformed into a forecasting technique. This is not to say that it may not actually be useful and have a positive impact on statistical techniques.

Keilis-Borok: To your question, I would give the same answer. The social sciences cover such a broad area that some problems probably require the perspective of chaos, but certainly not all. So it depends on the specific problem. But I object to what you are saying. English is not my mother tongue, but I do not think that prediction means just prediction in terms of the weather. I think the ultimate test of science is to deduce the existence of some phenomenon that was not known before and then make tests with data to see whether it is really observed. In this sense, prediction is a very crucial element.

I received a violently negative response to my paper on a recent prediction about the US presidential election, and the Nobel Prizewinner who reviewed it thought: "What do I know from reading this paper that I did not know earlier?" But he accepted the paper after we had provided a comment of a couple of paragraphs addressing that point. So in that sense, I think prediction is a requirement. Good science should tell you something that you did not know before.

Boldrin: You got me there. I was interpreting prediction in terms of numerical prediction - in terms of forecasting. So your English is probably better than mine.

Mandell: I want to say three things. The first is from the experience of Robert Devaney and Heinz-Otto Peitgen, both of whom are making a living these days dazzling people with computer graphic representations of dynamical systems. I have been in two audiences of high school and college graduates who gave them standing ovations for their talks. I just think that anything that creates such excitement about mathematics, particularly in the USA where it has become, as Celso Grebogi says, "a hateful subject," is likely to pay off in ways that may be totally unrelated to the actual content. So I think chaos is undeniably both educational and energizing, at least in the United States.

The second thing I want to comment on regards theoretical work. It might have come about anyway, but the way of working in this field is interactive, experimental, and intimately involved with the computer. It occurs to me that experimental mathematics, if that is a field, certainly has used the chaos phenomenology to attempt to make tackling a problem much like composing music by sitting in front of the piano to see how it sounds first. So there is an intimate interactive relationship with the computer that can result from the application of chaos theory.

And thirdly, I think chaos is used to create computer-graphic mathematical objects that are neither properly physical nor purely mathematical, but have sort of mediating existences that various disciplines can commonly talk about. I would not be surprised if that aspect becomes an influence on mathematical styles of work as well. None of these three points really has any intimate relation to the substance of chaos but rather to the process of chaos and its impact on education, research, and styles of scientific work.

Shearer: It may be a semantic point, but I have a question in response to what Michele Boldrin said: to what extent is "chaos" an instrument and to what extent is it an element of reality that we are learning to understand better?

Feigenbaum: I largely agree with your comment. One can certainly say that chaos represents some new sort of surgical tool, but it is equally true that one is trying to understand, at any level possible, what some variety of phenomena look like. This really means doing mathematics, or maybe physics or biology. We are rather lucky that we have succeeded in figuring out one or two little things that we can use. I think it is rather clear that our success is due to a little measure of chance, for there remains such an immense number of problems for which we lack ways of understanding and tools. One only hopes to be able to go further eventually. So if chaos simply refers to the particular tools we have developed so far, and of course it must partly mean that, then certainly chaos sits as a tool on a shelf to be used. But it is equally true that the thinking that led to the work done to develop chaos did not produce a final end product for a specific use that was so conceived at the time of development. Chaos has its own scientific perspective and it delineates certain problems, methods of thinking, and how one conceives of mathematics in terms of serving other intuitions. So that is also part of my understanding.

Ruelle: I would like to say that physicists do not like to speak about reality. They handle idealizations of reality and there is quite a choice for such idealizations. Some are fruitful; some are not. And the idealization of chaos or dynamical systems has turned out to be surprisingly useful.

Mayer-Kress: One remark I want to make is that one of the contributions of chaos theory occurred mainly as the result of a coincidence with the appearance of computer technology and interest in computer modelling. Now, we have not only mathematical tools for dealing with complex phenomena, but we can also perceive the results. If we would have had only the results of computer models available as a string of numbers, then no one would really be able to perceive the results and the advent of chaos would not have had the public response that it has today. Now we are able to discuss complex phenomena in a language that everyone understands, where we can just talk in terms of images and global relationships that are not formal and, therefore, are not restricted to the more mathematically educated population.

The second comment that I want to make is that chaos has had an impact on society to a certain degree, which is in opposition to religious beliefs and the belief that science is exact and therefore also applicable to fields where such exactness is motivated by some interest, perhaps economic. I think in many fields it is as important to show the limitations of predictions and the accuracy of possible predictions as it is to make the prediction itself so that you can have a feel for the implications of decisions in the context of those complex phenomena that really limit the predictions and put severe limitations on their accuracy.

Bishop: I would like to ask a question with implications perhaps for the future. So far in this conference, we have discussed details of low-dimensional chaos and particularly the embedding of low-dimensional chaos in other systems where we are not exactly sure what is going on. I think this raises some questions. What happens if there is no underlying determinism that we know of? Under such circumstances, can we do such embedding? If we are dealing with a more complex system, how and when can we reduce it from a higher-dimensional to a lower-dimensional system? I would like to know what things are important for us as scientists to consider in studying and applying chaos.

Glass: I will begin by making some comments which reiterate some points that David Ruelle mentioned about biology and then I will also say a couple of things about physics. Physiological systems and biological systems show complex rhythms at all levels, sub-cellular, cellular, and physiological - that is, in systems at any organismic level and in collections of organisms. This is just a fact about these systems. The approach that our group has largely taken, and also the approach that was beautifully described by Kazuyuki Aihara, is to attack such problems using the standard approach of the methodology of researchers trained in physics. Thus, we propose theoretical models, undertake experiments to test the theoretical models, adjust the models, and pursue this interaction between theory and experiment. As it turns out, the appropriate language for these theoretical models is the language of nonlinear dynamics and frequently of chaotic dynamics.

There is another set of research approaches that has not been discussed very much at this conference, but that perhaps should have been better represented. In this research, scientists are using tools that have been developed in nonlinear dynamics to describe biological systems as well. Brief references to this have been made a couple of times. Researchers are computing dimensions and often use a power spectrum, entropy, and the variance of these to describe biological systems.

I believe that the algorithms that are being applied are extraordinarily subtle and that not all practitioners applying these algorithms understand the subtleties and difficulties associated with them. As a consequence of that, we often find very striking claims in the national media including the press that such and such a biological system has been found to be or not to be chaotic. I believe that one must take all such claims with a grain of salt. Behaviours are complex and do not necessarily exhibit deterministic chaos, even though one reads that some distinguished expert has made that claim. This is not to say that researchers should not use and study these tools. Of course they should. But I believe scientists must be conservative about interpreting the results of such applications and studies. At the moment, there is no impact of chaos on medicine. In the future, that is within the next one or two years if not sooner, there will be dramatic claims that chaotic dynamics and tools introduced from chaotic dynamics have important prognostic significance in medicine. I believe there will be claims - I have three friends who make these claims now - that various dynamical characteristics will be able to predict who has a high risk of becoming a victim of sudden cardiac death.

Some of these claims will be driven by the profit motive. There is an enormous amount of money at stake in institutionalized medicine in the United States, Canada, and most of the industrialized western countries. When Peter Grassberger and Itamar Procaccia developed their algorithms, I do not know if they thought about peddling them. At the moment, the situation is quite different for researchers studying biology. There are many scientists now who are aggressively patenting algorithms for computing dimensions, even for such things as power spectra and auto-correlation functions, and who are trying to aggressively pursue their patent claims. When researchers begin claiming that they have techniques that have predictive value in medicine, it is very important that there be experts around to ask them for details of their algorithm so that it can be critically tested. This is not to say that the researchers will purposely mislead but that the basis of scientific development is independent investigation and testing.

At this conference, I think that there has been a striking absence of experimental observations. This is not as true for biology as it is for the physical sciences. I do not mean to say that there is not a large body of beautiful experimental work that has been done in physics, but rather that I think it really has not been presented at this particular conference. One possibility for this situation is that there is a definite prejudice among physicists, particularly among theoretical physicists and I would also say especially among Japanese theoretical physicists, that one should be working only on theory. And theory in nonlinear dynamics is very often dissociated from experiment.

Why is this true? One of the reasons has been expressed by Mitchell Feigenbaum: physicists are interested in universality. Once I have a theory of universality, I do not necessarily have to look at an individual system because my theory is going to be good for "all" individual systems, or more precisely, a great many individual systems. I believe that to be correct to an extent, and it is a very beautiful observation that the same phenomenon is observed in many systems. However, I believe it is absolutely essential to look into the details of individual cases also, because it turns out that there may be many subtle features of a universal nature occurring, but only in one little corner of parameter space. All throughout the rest of parameter space there are many other things happening for which we do not have very good theories because they are not yet known to be universal. So I think that it is essential to look in detail at a variety of specific systems. In physics, when researchers look in detail at specific systems, for example the absolutely fantastic work done by such scientists as Albert Libchaber, Harry Swinney, and many others, actual experiments with these systems are often of interest because they serve to exemplify certain points of universality in the mathematical models, rather than being of detailed interest in that particular discipline.

In biology, however, I would say that the situation is somewhat different. Examples and experiments involving biological systems have been and will continue to be of interest from the point of view of the actual discipline being studied. The properties of the arrhythmicity of beating hearts were looked at long before I or any other mathematically oriented researcher had become interested in them, because this is a matter of extreme importance to this particular discipline. So I believe that nonlinear dynamics and chaotic dynamics have a natural home in biology merely because, due to the structure of biology, equations do not have the same sacred character that they have in physics. In biology, no one has a Schrödinger equation or a set of Maxwell equations. One is always in some topological space of equations and, in the equations, there is an absolute essential necessity for a topological perspective towards the dynamics.

Finally, I believe that there is a compelling need for mathematicians and physicists to think in detail about problems in biology. This necessarily implies that there will be interdisciplinary work. It is essential that researchers avoid doing physics and mathematics while sitting in their ivory towers. They have to actually see the experiments. They have to see what is happening in hospitals. There is an essential need for interdisciplinary work and at the moment this is extremely difficult to organize.

One reason is that there are very few natural ways to support such research and even when scientists take the step actually to strike out on this pathway and do such research, there is no guarantee of a future job. In fact, which physicist who is now running a department is willing to give a job to a young physicist who has been studying cardiac electro-physiology, ecology, or the properties of firing channels? So I think that for any people involved in science policy-making, some understanding of biology is absolutely essential and thus some serious thought should be given to how funding, both in terms of research money and in terms of future physicians, can be directed in such a way as to provide employment to young people who are interested in working in these areas.

Feigenbaum: If one examines where there have been successes and serious impacts in physics from chaos, as David Ruelle has already mentioned, the subject of fluid turbulence comes to mind, a subject that is no longer the same. It is regrettable that we do not have any experimenters present with respect to this field because, as a matter of fact, not only has the subject of fluid turbulence been seriously modified, but also it has been modified to the point that it now consists of new parts. So the onset of turbulence is now a much larger subject than existed some 10 or 20 years ago and, of course, one knows that the theoretical framework is now completely different from the old field theories, with their Gaussian forcing terms, which when graphed produce uninteresting results. We certainly do not have the tools to generally understand very complicated turbulent behaviour, which remains an important problem. We are not certain if any of the concepts of dynamical systems will give us the ability to understand other truly chaotic behaviour, such as fully turbulent fluids. We might not have the right set of ideas. When you see some of these truly extraordinary experiments, the mode of operation is also rather different from what was done in the past. One now analyses data, for example, with an eye toward finding dimensions.

Other methodologies are coming out of chaotic thinking that are part and parcel of the standard things one does now but never saw 10 or 15 years ago. These are all modern things that are clearly the place where chaos has made an impact in physics. Again, as mentioned by Leon Glass, there is the work of Harry Swinney, perhaps John Hudson, and a whole host of researchers, including a large French group, tackling the general question of what happens in oscillating chemical reactions. Our knowledge of dynamical chemical reactions is absolutely transformed from what we understood before. To put things in a little better perspective, it slowly came to be understood some 15 or 20 years ago that you can have regular oscillations in some organic chemical systems. But when it became rather clear from very carefully performed experiments that one also observed chaotic behaviour, some of the great masters of the subject, for example, Morris, categorically denied that it was possible that there could be any chaotic behaviour in those systems, because according to him, it would violate thermodynamic laws. The result was, of course, a subject that evolved into something very different from what it had been. We now can analyse, to varying degrees, through this general dynamical systems way of thinking what sort of reactions can been seen and what kinds of dynamical processes can occur. Of course, this general notion of oscillatory chemical behaviour will have, at some point in the future, various biological applications. So I think I absolutely concur with Leon Glass. It is regrettable that we do not have an experimental chemist or experimental fluid physicist who could indicate a little more directly what some of the methodologies have become, because indeed they are very different from what was used in the past.

Wisdom: Considering what you just said about the irreversible and long-term impact, both current and future, of chaos on all these areas, I am curious about your statement, which I have heard before, that chaos might be like catastrophe theory, that is, it might be something that is going to pass out of fashion. Such a statement about chaos sounds totally ridiculous. It is obvious that chaos has fundamentally changed how we look at things.

Feigenbaum: Catastrophe theory, purely for reasons of sociology, I suppose, suffered from an initial hard sell followed by an excessively strong counter-reaction. Catastrophe theory has at its heart a rather potent theorem that in certain circumstances allows you to make some penetrating observations. To my mind, the very best thing that was ever accomplished using catastrophe theory was by Michael Berry in understanding the general properties of caustic propagation. Thus, one can analyse some truly interesting problems based on the results of this theorem. It should be said, however, that its domain of applicability is rather small. It is clear, on the other hand, that the ideas of chaos in dynamical systems theory embrace a much broader range of things and we are simply coming to realize that we can see much more complicated behaviours in these systems and that we have some tools to quantify them. Once seen, these behaviours are not easily forgotten. They were not seen in the past because they were considered noisy perturbations to be thrown out.

Using these methodologies, we can only understand behaviour of a rather limited number of degrees of freedom. If the ways we think about these phenomena, our images of them, and the way we find relations between things such as the seeking of dimensions were enough and led to the full set of tools needed to analyse all problems, then our capabilities would be all-embracing. Unfortunately, we cannot stand up to a serious fluid engineer and claim that we know how fluids work. That would be completely ludicrous; we have no such capabilities. We do not understand the mathematics at the moment and, although we can understand pictorially that we probably have a rather good idea as to what is going on, we do not at all have the tools to handle this problem. If this line of thinking does not lead to those tools, then, of course, other approaches will be needed to deal with those problems. One can ask: have we really milked this approach for all we can? I think the answer is "no." I think Leon Glass would probably say the answer is "no" also.

Glass: I think that we hope to discover new things, but I would also like to quote Stan Ulam, who said: "Ask not what mathematics can do for biology, but ask what biology can do for mathematics." I would like to believe that, already at this stage, the focus or some of the focuses resulting from thinking about biological problems have already had some effect on the choices of problems that people have tackled in mathematics and contributed to some of the directions that mathematics has taken. I think that this is something that we can look forward to. I do not believe that, in the work that I do, it is simply a matter of looking up the paper of someone who has proven some theorem or obtained some good result from applying it directly. I think that novel problems are being formulated based on what researchers are finding in biology, which also constitutes a significant aspect of the work.

Grebogi: I would like to make a comment about the matter of experimentalists at this meeting because this issue was brought up so strongly. I would like to point out that I spent one-third of the time of my presentation talking about experiments, Steve Bishop talked about experimental work concerning ships that capsize, Michele Boldrin gave experimental results, and Vladimir Keilis-Borok spent his entire presentation talking about experiments. I also provided an example in this discussion about the space shuttle that concerns an experiment and is reported in an experimental paper.

Mandell: I want to defend catastrophe theory, not as practised by Christopher Zeeman or reported by the newspapers, but as a mathematical analogue that prepared us to anticipate and accept the fact that, from qualitative behaviour, there are finite sets of solutions that began with catastrophe theory and that can be applied to practical problems using applications derived from this very abstract type of mathematics. For good or bad, I think the problem it encountered was sociological and not mathematical.

Feigenbaum: There is a limited spectrum of problems to which one can apply catastrophe theory. This range is very sharply defined and beyond it catastrophe theory should not be applied.

Mandell: No, I am not saying that. I am saying that God is ready to accept the idea of universality. I think it is quite amazing that, just given dimension, leading researchers have extracted so much information about singularities. And I think that in the flow of development of mathematics, the value of catastrophe theory cannot be dismissed just because it was oversold and cheapened. Catastrophe theory involves a transfer of very pure mathematical ideas to applied work. If a danger of misunderstanding existed, it was not because of the superficiality of catastrophe theory, but the grand idea of selling its applications. In that sense, it serves as a very good lesson about what not to do. So I am not entirely sure that you should throw out the baby with the bath water, as far as singularity theory is concerned. I think physicists in particular have used catastrophe theory as an example of an application of mathematics that got out of control. Mathematicians do not like this because they feel that their work was cheapened as a result. In saying this, I am not defending the application of catastrophe theory to problems in social science, economics, and personality theory. But I do think it served as the necessary analogue for this kind of universality concept.

Feigenbaum: For whatever it is worth, I think that where catastrophe theory fell awry with physicists is, of course, that it is a classification of phenomena of a specific number of dimensions. It ran into questions of phases and of thermodynamic materials. The important question of why physicists paid very little attention to statements about catastrophe theory is that, to a first approximation, physicists have no interest in any problem that does not have an infinite number of degrees of freedom. They are fundamentally not interested in catastrophe theory. That is a community decision and the rejection of all of the ideas of catastrophe theory, if one listens to what Michael Fisher has to say.

There is the separate issue, of course, regarding the overselling of sociological applications and I do not feel confident to say very much about that except that one should have been sceptical about it. Certainly we have seen one or two truly beautiful applications in physics. One is the understanding of the speckles and jewels of light we see at the bottom of a pool - that was an absolute tour de force.

Takeuchi: Let me first begin with why I became interested in nonlinear models and especially chaos. As a statistician and also an econometrician, I have played around with many kinds of econometric models - models that represent the economic system by a number of equations from which we can estimate the parameters and then predict or analyse actual data. This approach has been widely used for 30 or 40 years and in almost all cases the models are essentially linear. Some of the models are allegedly nonlinear and we have to apply techniques to deal with them as such. This is the reason I was unsatisfied with papers representing the status of the art of econometrics, although it often seems to me that nonlinear models are not satisfactory either.

The second reason for my interest concerns the theoretical aspects of economics. Mathematical economists have long been interested in proving the existence and the centre of equilibrium of economic systems. They are particularly interested in a unique, stable equilibrium and so they have tried hard to prove that the existing equilibrium is stable and hopefully unique. They have tried very hard to find out the conditions under which a unique, stable equilibrium can be established in some very large economic systems. They are not usually interested in systems where equilibrium does not exist, is unstable, or is not unique. This approach seemed to me to be a bit too idealistic and not relevant to real economic systems. My interest was in developing a little more satisfactory theory dealing with the nonlinear aspects of economic systems either from an empirical viewpoint or from a theoretical viewpoint.

Even among the social scientists and economists present here, there are widely varied interests such that what they have in mind as mathematical models of a social system may be quite different from one to another. So I would like to be a little more precise about what I have in mind as a typical model for an economic system. I am not interested in stock market models. I am interested in what are called multisector dynamic macroeconomic models.

For example, consider some national economy such as the economy of Japan. That economy consists of many industries: primary industries, secondary industries, tertiary industries, agriculture, manufacturing, construction, services, etc. Each industry produces goods every year using labour and capital, and also intermediate goods that are the product of other industries, such as material and equipment. What is produced every year is either consumed by consumers and/or used for further production either of materials or as capital stock, such as machines and equipment. All of this can be expressed in a system of equations in a dynamic way such that this year's decision affects next year's decision in a variety of ways. In this way, we can formulate a kind of macro-dynamic multisector model of an economy.

Two types of equation are actually involved in this type of model. One is called the technological equation, which involves a relationship between countries that is technologically determined as, for example, when the production of automobiles requires a specific kind of steel. The other is called the behaviour equation that deals with consumer response to the relative price of each type of consumer good under, of course, the constraint of the consumer's available income. Also there are several types of institutional equation, for example, that involve the government collecting taxes. An assemblage of these equations can be used to represent a dynamic system. What would be the dynamic nature of such systems? When we consider the technological equations and behaviour equations at the same time, there are actually very subtle and delicate aspects involved. When we consider the actions of each sector responding to the different changes of relative price, we find that the system becomes, in a sense, very unstable. Not necessarily always unstable, but very sensitive to changes in relative prices, because if one sector is losing money then production in that sector contracts. But if that sector makes a significant profit, then investments would be made in that sector and production would grow.

What interests me is that this type of dynamic economic model has, thus far, not been satisfactorily analysed theoretically. The most successful model is the balanced-cross model, which is theoretically all right but very unrealistic in that all sectors grow proportionally at the same rate. My interest is in knowing whether this system is really a type of nonlinear dynamic system that might possibly be described as chaotic. Any mathematical theory in the social sciences is admittedly only a crude approximation to reality, so I do not think it is possible to achieve a full description or long-range prediction based on mathematical models. All that we can do is develop a rather crude prediction or understanding of the mechanisms of economics, and other social systems.

Can chaos theory provide us with such a deeper understanding of the economic system as it actually works? To me it seems that there is hope that chaos theory can help us better to understand economic systems and especially illuminate some of their very intriguing and subtle aspects. We know that economic systems are sometimes stable and sometimes unstable, but often have a very strong power of recovery from apparently desperate situations. So my interest is in how and in what situations this phenomenon can happen and I hope that chaos theory can explain these things using some kind of dynamic systems model.

I am also interested in the prospect of using nonlinear models to compare different types of system with the hope of finding similarities among very different models. For example, some of the ecological models I found have features very close to some of the socio-economic models. I would also like to be able to incorporate into these dynamic social models aspects of the condition of the natural environment and natural resources, especially thost which are renewable.

Feigenbaum: I would like to make one or two comments about where I would like to see things move. We should bear in mind that, with this mathematics related to chaos, the number of situations that we can understand at the moment is truly very limited. If we look at bifurcations of co-dimensions of any modest number bigger than about two, we have difficulty dealing with them and the immense number of details that begin to appear. Therefore we might wonder if this is the right path along which to proceed, for certainly we would like to see more progress in understanding these phenomena. If we try to examine and understand the general circumstances of the kind of scaling structure that give power laws - the types of analysis we know how to do - we see that we can do rather well on the basis of functions of one variable. We know how to inject them and imbed them into much higher dimensional spaces but, nevertheless, I would say that we have made surprisingly little progress in going much further than that over the past five years. This semi-analysis was done in functions of two dimension giving us a special geometry to understand things. But the mathematical landscape of a large fraction of this type of problem that we like to understand is still very elusive and it remains to be seen how much progress we can make.

Already in the context of looking at some inflective two-dimensional behaviours, we immediately realize that our understanding of dimension is not very good. While it has many things that are useful about it, dimension is a notion that is completely incompatible with any scaling properties and it is rather clear that we need a much better definition of dimension. Many of the applications and data manipulations that we think about have very much to do with determining dimension: how many different components one is looking at that actually give rise to the data. One of the best algorithms that exists is, of course, that of Peter Grassberger and Itamar Procaccia, but that algorithm is not very good. We would like to see much better algorithms coming into existence.

In biology, we immediately recognize the shortcomings of our techniques. One of these shortcomings is the need for very long data streams. In a very nice analysis, David Ruelle more or less says that you should use the logarithm to the base of 10 of the number of points. That means that if you have 10⁶ points, you get 6. This gives a very good idea of what the upper bounds should be and is an indication that you can believe anything coming out of such an analysis. This is a profound limitation for biological measurements because we do not have that long a data stream. We should contemplate that, if there are multiple channels present, and thus larger amounts of parallel data available, can these be used over short periods of time to obtain good analyses? That would be of decided interest to a variety of biological applications. From a survey of what we know and what we use as tools in chaos, it is clear that there is a very large amount of open territory. The question again is: do we have any ideas as to how to deal with that territory?

Bishop: I wonder if anyone might like to suggest whether the most significant questions are still in chaotic theory and particularly indicate the implications they might have in other disciplines.

Ruelle: It is not true that the application of dynamical systems theory to physical systems is limited to small dimensions. It depends very much on whether you have control over the equations of the system. So, for instance, in meteorology, one has a good understanding of the basic equations and therefore one can handle systems that have dimensions that are by no means small. The same thing is quite true for astronomy where we have an extremely good understanding of the basic equations and, therefore, we can do things that would be quite impossible if we had only experimental observations to deal with. So, progress has been made in applying dynamical systems theory to physical systems with results where we are better able to control the equations of these systems. That is not an unreasonable proposition. What you have is a system that is complex and that you do not understand very well. By treating relatively short time series, you get a better notion about what the basic equations are and, when you are confident that you understand the basic equations well, then you can put them into a computer and analyse them. That was done, in a modest way, with oscillations where we still do not understand basically what is happening. By analysing these oscillations we now have a model dynamical system that reproduces the behaviour of the oscillations and permits predictions.

Mayer-Kress: I think one point, that has to my knowledge not been discussed here so far, is that you can also use observations and reconstructions of models for chaotic systems to influence the system, and then look at the response of the system to the impact from the model and see how well the model is doing in controlling the system. This entire domain of using chaotic dynamics for controlling complex systems is something that I think has a lot of potential in the future and is currently being studied, especially at the University of Illinois.

Glass: The question was "what is needed in chaotic systems?" In the years in which I worked in biology, there was one thing that would have been of enormous value. It would be extremely nice if one had an operational test to determine if a system is deterministic or not. That would save everyone a whole lot of time, but this is a different question from seeking its dimension or power spectrum.

Feigenbaum: Are you sure that is a different problem? If you were to get a very clean power law that says the dimension is 3.7, that would solve the problem.

Glass: That may be, but anyone who has ever tried that in biology rarely gets very clean power laws. They get 3.7 but not over the same order of magnitude of scaling regions that you can get with a Lorenz or some other system with 200,000 points to 10^-14 accuracy.

Feigenbaum: I completely concur with that. That is what I was saying about using parallel data streams. It is very clear from looking at biological applications that one needs better quality data.

Bishop: I concur with that. The data are never clean when it comes from biological systems, which makes it very difficult.

Takeuchi: In economics, the data may be even dirtier than in biology. And so I think that, in economics and other social sciences, we can formulate a variety of models based on the same set of data and thus there is a wide range of choices for the form of these models. So I think that even for nonlinear models we can choose either a non-chaotic model or a chaotic one. So, assuming that the choice is not forced on us by the kind of data set but rather that we can choose to use chaotic models, I think that the advantage of chaotic models is that they provide a deeper and more useful understanding of the economic system and how it works. One of the reasons is political and involves policy-making implications, such that some of the policies will lead to very different results in different situations. Some policies may be beneficial and profitable, and some may not. In some cases, a slight change in policy could prove disastrous and such a phenomenon could, I think, be interpreted using a chaos model.

Aihara: I have some comments from the viewpoint of neuro-physiology. The brain is a system of very high dimension with a huge number of degrees of freedom and, moreover, it contains very complex interactions. Looking at individual interactions, we can analyse their dynamics by taking a manageable element of very low dimension. In order to understand the brain, we must analyse a behavioural network composed of 10 billion such chaotic elements or subsystems. Special tools are needed to analyse a system of such high dimension.

Feigenbaum: There is no question about that. Of course, it could turn out that the 10 billion-element system has in effect a dimension that is much lower than 10 billion, and certainly will be smaller than the total number of neurons, but nevertheless an impressive problem remains. The question becomes whether we can refine the tools that we have, or, if this whole set of ideas is inappropriate, whether we can learn to develop new tools.

Boldrin: In his statement, Yoichiro Takeuchi seemed to imply that, in economic policy analysis, the use of nonlinear analytical techniques as opposed to linear ones may show that the same policies have different effects and different outcomes. I would like to have an example of this because that is one of the things I have never found in any model we have been dealing with. The effect of one policy or another typically depends on the structure of the model and on the economic assumptions made in building it, but never on the kind of analytical technique or formal description. So, if you have an example, I would be very interested to see it.

Takeuchi: I was not quoting any example that I have observed, but I think you will agree with me that similar forces sometimes produce quite different effects under different situations. So if it is possible to explain these divergent results using one model applied to the different situations, it would help us in understanding the economic system and how economic forces work.

Kaneko: Leon Glass commented about the shortcomings of physicists, but I think it relates primarily to western physicists. The reason I say this is that the universality Mitchell Feigenbaum mentioned is only quantitative universality. This limits application only to such concepts as turbulence. But eastern physicists can deal with complex systems such as complexity, and therefore can study qualitative universality in truly complex systems. That is what is needed when we try to go to chaotic systems of a very high dimension, such as biological systems.

Shearer: Can anyone think of practical applications of chaos for the near future, aside from those described in the medical area? Are there others that we can expect in a practical sense?

I also have a more philosophical question. We are discussing deterministic chaos. But determinism is basically a reductionist concept and chaos is more of a holistic concept. Do we perceive the advances of chaos as strengthening the reductionist approach that will allow us to treat a wider range of phenomena more effectively? Or is the study of chaos ultimately going to lead to the strengthening of the holistic approach that will help us to deal more directly with complexity?

Peixoto: As for the immediate applications, they are being developed, but not commercially. They occur more in industry and shipyards.

Bishop: That is right. University people do not tend to be so commercial, but there are many commercial aspects that have been designed and will be conceived in the future. Most of the work in this direction that we have seen so far deals more with chaos avoidance rather than the actual usage of chaos. But I think there will be more applications that utilize chaos in the future.

Grebogi: I think I already mentioned a couple of applications. What about the mixing problem? It has been studied quite effectively in the chemical industry. If you want the reactants to react effectively, you have to ensure effective mixing, and chaos provides that because of the stretching and folding. That is one problem that has been studied quite a lot and to which chaos has been applied effectively.

I will also mention the old example involving thermonuclear fusion. If microwave energy is introduced from outside the plasma, the microwaves are going to dump their energy in a particular location of the plasma, whereas what you want is to heat this plasma uniformly. Therefore, you want to add chaotic properties to the radiation.

Another application that we expect to be important in the future is the control of chaos and I think the way to do this is to use the chaotic attractor to gain access to very large regions of parameter space. When you design a system like a chemical reactor for fixed values of a parameter, you can stabilize different regions of this space.

Takeuchi: My philosophic question is: what is the relation between chaotic theory and stochastic theory? Can there be stochastic chaos, that is, behaviour governed by stochastic rules and also combined with chaotic properties? Is there any possibility of that?

Ruelle: I think this is to some extent a semantic problem. A stochastic variable is one that varies in a random way, according to a certain probability distribution. You can create that using a dynamical system that is chaotic. One is a special case of the other.

Mandell: There's a mathematician, David Knowshim, in Bremen, Germany, I believe in the school of Ludwig Arnold, who is working right now on stochastic differential equations in the context of dynamical systems and uses enough exponents so that there is a marriage of the formalisms.

Speranza: I think it is necessary now to mention the fact that the theory of stochastic perturbed dynamical systems was published in a paper by Haken in 1972. Probably, what you are alluding to is whether the noise is considered to be external or internal. This is not a semantic question. You use stochastic noise in order to reproduce a certain kind of statistics that could be produced in a larger system by a collection of modes that would be, in that case, internal to the system.

Takeuchi: I am referring to the kind of internal uncertainty or unpredictability as formalized in some stochastic processes. To me, it seems that chaos is a concept that is rather stochastic.

Speranw. At any rate, we have a decent theory of stochastically perturbed dynamical systems available. We can work upon that and sometimes it proves to be a preliminary step on the way to obtaining the same statistics.

Feigenbaum: I think this session has come to a close. Thank you all for participating.