When my father heard that I had added an interest in chaotic phenomena to my previous research into catastrophe theory, he remarked that mathematics was putting in a bid for the title of ‘the dismal science’.
Indeed the names of these two new but exciting branches of mathematics are somewhat off-putting, and one might be pardoned for concluding that gloom and despond are appropriate states of mind for a mathematician with such interests. This, fortunately, is not entirely the case.
In fact, as one looks at the two theories, one is struck by a profound difference of emphasis or, if you like, of mood. Catastrophe theory, belying its name, is optimistic in this sense at least: that it offers the hope of a mathematics dealing with discontinuous phenomena, processes exhibiting sudden transition or violent change. It promises to extend the scope of mathematics, as the traditional methodology (calculus) is inapplicable in such cases. Chaos theory, by contrast, begins with a pessimistic insight: there are, quite possibly, natural systems about which we can say very little, whose behaviour we cannot predict and whose basic laws we can never know.
Mathematics, as such, has no direct bearing on the world of experience. It discusses objects that, whatever metaphysical status we ascribe to them (and opinions vary here), certainly have no real existence. One doesn’t walk down the street and meet a correlation coefficient; no instrument will observe a partial derivative. In other words, mathematics can say nothing, in a direct way, about the world. But it does not follow from this that it is irrelevant to the view we take of our surroundings.
Applied mathematics is alive and well. The traditional, and necessary, procedure adopted when mathematics is used in a real life context is the technique known as ‘modelling’. The name derives from the more familiar use of the term as when (say) a cat is modelled in clay. In the mathematical case, a problem in, for example, navigation, is translated into (or modelled by) a question of pure geometry. There are three components to this process. First, the basic problem is restated in mathematical terms; second, the mathematical question is resolved; finally, a reverse translation produces, it is hoped, a usable result. There is an analogy (not by any means a perfect one) with the model cat. The real cat is observed, some essence of its felinity is imposed on the day, and the result is appreciated.
So automatic does the modelling process become that navigators, engineers, physicists, astronomers and all the other traditional users of mathematics come to do it quite unconsciously, without realising even that a translation process is involved. A speed is v, an angle a, a temperature T, a pressure p. We tend to forget that these symbols represent measurements that relate in quite complicated ways to the underlying concepts. Usually, it doesn’t matter that we do. For in these traditional areas of application, the translations are accurate enough to sustain elaborate constructions, verifiable and useful in practice.
Our navigator, taking a star-shot, need not be troubled by the thought that stars are not geometrical points but fiery balls of plasma rushing headlong through galactic space, that he may be sighting a star that blew itself up millennia ago, that this very star still exerts a gravitational force on his ship, itself a tenuous structure of nucleons and electrons. The minute effects of cosmic ray bombardment or radioactive decay do not concern him.
Nevertheless, every translation into mathematical language involves some inexactitude, some idealisation, and other situations are not so clear. The applied mathematician seeks to preserve some factors while he ignores others. The success of the enterprise depends very much on the relative importance of the different components. It is typical of the biological and the social sciences that modelling in these areas is trickier and less precise than in Physics and its offshoots, where small numbers of predominating factors are more readily identified.
This century has seen persistent, and by no means entirely successful, attempts to mathematicise the soft sciences. One of the happiest examples has been the development of an extensive, elaborate, and at times quite useful science of quantitative genetics. Yet even this can only succeed in a very limited sense.
Each individual human possesses some five million different genes. Although other organisms are simpler, none is so simple that it carries only six, and this case is too complicated already for a full analysis. Even the case of four genes is unresolved in many of its details. The complete account of two-gene ‘organisms’ (genes tend to come in pairs) is less than twenty years old.
Add to this a further difficulty — that genetic phenomena depend upon environmental and even social variables — and the problem becomes even more formidable. Beyond this difficulty lie even more troubles, for the different effects interact in unknown ways.
Mathematical models do exist in the socio-biological sciences, and they work. That is to say, they work in limited areas. The necessary simplifications of the modelling process are so crass, so extreme, that they cannot sustain an elaborate superstructure of theorem and subsequent theorem. In biology, the modelling technique has been used with most success in the areas of genetics and ecology. Yet we still can say very little on the breeding of racehorses or the effects of pollution in Westernport Bay. There are too many variables and we don’t know how they interact.
Biomathematicians and others, finding themselves in these straits, have come to ask more fundamental questions. If I avowedly oversimplify a situation, what conclusions may I legitimately expect to remain valid? How drastically will imperfections in measurement affect the conclusions I draw? What if the basic laws of the system are only known (or knowable, even) imperfectly? Can we say anything of cases where what laws exist do not lend themselves to the traditional formalisms?
It is in this area of metatheory that the terms catastrophe and chaos arise. Significantly both developments have strong biological links. Catastrophe theory is pre-eminently related to embryology and chaos finds ready application in ecology.
Catastrophe theory is the brainchild of the French topologist, René Thom. (Topology is a very general form of geometry.) Thom, in 1958, won the Fields medal, mathematics’ equivalent of the Nobel prize, for his definitive solution of the ‘cobordism problem’, whose details are fascinating but need not concern us here. He is a man of many parts, a skilled lecturer, an able publicist, an amateur linguist and philosopher, and a highly gifted and brilliantly intuitive mathematician. Between 1961 and 1966 he directed his attention first to optics and later to embryology.
Deep underlying similarities between the two fields led him to enunciate his now famous ‘Theorem of the Seven’ — the basis for the new theory. According to this theorem, all mathematical functions, provided they satisfy certain plausible constraints, are reducible to seven basic archetypes. Each archetype corresponds to a mode of sudden change: a continuously varying cause producing a sudden discontinuity of effect.
The cause or input is supposed to consist of one, two, three or four ‘control variables’, and the state of the system is then shown to depend upon at most two further variables (‘state variables’). Providing only that the state of the system can be given in terms of a single function (a condition often, but not always, true), and that this function involves no more than four control variables, any unusual behaviour will necessarily conform to one of the seven archetypes. These are referred to as ‘catastrophes’ — not that they are necessarily disastrous, but rather because there is no better English word to convey the nuances of the French catastrophe.[i]
It is this result that gives rise to the amazing breadth of applicability claimed by the theory. Analogies between apparently disparate areas of science are seen to be the reflections of fundamental structural identity. We are thus able to see why the buckling of certain types of strut should be subject to the same equation as the liquefaction of gases. With two control variables involved, in each case, both must conform to the same unique archetype.
This same archetype describes patterns formed by light-rays, the behaviour of various engineering demonstration devices, and the action of the heartbeat. More speculatively, it has been applied to prison riots, cell division, manic-depressive states, animal behaviour and curricular fashions, to name but a few of the studies so far published.
Thom’s early applications dealt in most detail with embryology, a fact which even influenced the title of his book.[ii] Nowadays, he addresses himself more to linguistic applications, of which the most developed and probably the most successful is a classification of verbs.[iii] E.C. Zeeman, of the University of Warwick, is the best-known of Thom’s disciples and a most prolific author in the field. Zeeman espouses the view that catastrophe theory is the pre-eminently ‘right’ branch of mathematics for the sociobiological sciences and he has developed a large number of mathematical models in support of his claim.[iv] Many other researchers and schools are involved in the area, including an influential one at the Universität Regensburg, which produced the first published proof of the theorem of the seven.[v]
Thom’s theorem gives researchers a ‘handle’ on problems involving sudden or irreversible change, on problems where two extremes exist without any intervening middle ground. It is a qualitative, rather than a quantitative theory. That is to say, it is less concerned with numerical predictions than with the answer to the general question, ‘What can happen?’. For many sociobiological applications this is all to the good, for the kinds of question that crop up in these areas are also qualitative in character.
The limitation to two state variables is interesting and important. Given four (or, as subsequent research has shown, even six) control variables, no more than two state variables can be involved. At first sight, this seems to be absurd, for, especially in the social and biological sciences, experimenters are all too painfully aware that large numbers of variables play a part. What the theory assures the experimenter is this: that, whenever something interesting or unusual happens, only one or two of these participate. Unfortunately it doesn’t tell him which one or two.
Another advantage of the theory (one springing directly from its qualitative character) is that it is unnecessary to be absolutely precise in specifying the units of measurement. In some applications for example, ‘hunger’ is used as a control variable. That we might well have different scales on which to measure hunger can bedevil some discussions, but will not upset a catastrophe theoretic account. As this problem besets a lot of more traditional work in the sociobiological sciences, the qualitative character of catastrophe theory promises to lead to real advance in these areas.
For such reasons as these, catastrophe theory has become a most fashionable area of study. It has attracted more popular attention than any advance since Norbert Wiener’s Cybernetics became known in the late 1940s. (The only other widely publicised mathematical advance this century was Gödel’s incompleteness theorem, proved in 1931.) Thom’s work has been written up not only in New Scientist[vi] and Scientific American[vii], but also in Newsweek[viii] and even in the National Times[ix].
Whether the theory is as widely applicable as claimed by some of its apologists cannot be known until we have more experience with it. The theorem of the seven was first stated in 1967, and no proof was published until 1974. Although some 200 or more papers on applications of the theory have now appeared[x], some are known to be wrong, others are highly speculative, and yet others strike me as trivial or imprecise.
Applications tend to fall into two groups. Some are quite exact because the relevant scientific laws are known. Others are more speculative in that the applicability of catastrophe theory must be postulated. The first type tend to fall within the physical sciences, the second in socio-biological areas.[xi] Some critics, such as Croll[xii], use this dichotomy to characterise all applications as either repetitively reformulative or impossibly imprecise. This view, though extreme, has some truth to it. Certainly many ofThom’s applications are clearly metaphors to aid our understanding rather than scientific accounts of the traditional kind. Whether they will be accepted by scientists is a matter for the future to decide.[xiii]
In catastrophe theory, mathematicians have a new tool. At present, they are wielding it a little indiscriminately. Undoubtedly this mood will pass. Catastrophe theory will remain a part of the mathematics of the future, and an important part at that. Whether it will rewrite the syllabuses in applied mathematics is not yet known.
If mathematicians are understandably euphoric over the possibilities of catastrophe theory, they are perturbed by the implications of chaos. This sets major limitations to the scope of applied mathematics.
Many mathematical models take the form of a set of differential equations. Indeed, it might be said that, since Newton, the differential equation model has become a paradigm for the exact sciences. It works like this. Let me specify the state of a system and let me also say how it will change. It should then be possible to predict the entire future behaviour of the system.
This is the dream that came to be parodied as the ‘clockwork universe’, and indeed, in its pure form, it is discredited. Heisenberg’s uncertainty principle shows that it can never be absolutely valid.
However, for many purposes, the small effects of quantum theory (of which the uncertainty principle is a part) are quite ignorable. In such cases, we revert to the Newtonian style formulation, with the mental reservation that it will not be absolutely precise.
If our system contains only two differential equations, it settles down in the long run either to a state of complete equilibrium or to one of exact periodicity. This nice result cannot be extended, unfortunately, to cover more complicated cases and, generally speaking, these are the ones we
need. The more complex behaviour is in fact not only possible but almost routine. However, a heartening result appeared in 1966. To appreciate it, we need to digress for a while.
Suppose Og hits Blog in the year 20,000 B.C. How does this affect us today? Well, it might turn out to be, if only we knew all the facts, vitally important. We could, with imagination, write a scenario in which this event changes the entire course of history. More likely, however, we are quite unaffected. If such events do affect us over vast spans of time, history is greatly limited in what it can tell us, for every day innumerable Og-Blog punchups occur, and who’s to know what might come of it?
On such a basis, some scientists[xiv] have proposed that good theories should satisfy a requirement of ‘asymptotic stability’ — that the long term behaviour should not be upset by trivial events. Small causes are to have small effects that die out with the passage of time. In 1966 G.R. Sell proved that if this requirement was imposed, then all systems attained either perfect equilibrium or precise periodicity in the long run.[xv]
For some reason, not quite clear, applications of this work have tended to concentrate on ecological problems. If Sell’s theorem is applicable to ecosystems, then we would have good grounds for saying that a ‘balance of nature’ exists. Some ecologists, notably, in Australia, Charles Birch, would say that there is no balance of nature in this sense.[xvi] That is to say, they reject asymptotic stability as a requirement of their chosen discipline.
This school of thought receives support from the recent discovery of chaotic phenomena, which can be most pointedly discussed in terms of difference equations. These are conceptually simpler than their differential cousins, but can give rise to more complicated behaviour.
Whereas a differential equation tries to follow events as they evolve in continuous time, the difference equation allows only certain fixed times of sampling. Governments, taking census every ten years, use in effect the difference equation approach. More to the point, there are ecological and other natural phenomena which demand it. The periodic cicadas of the U.S. corn belt are an often quoted example.[xvii]
Computers, which cannot solve differential equations, are routinely programmed to replace them with difference equations, which do lie within their powers. Meteorologists who cannot, perforce, collect data from all over the globe, depend ultimately on a limited number of sampling points (weather stations) and thus on difference equations. Given these facts, the current practice of taking data from the network of weather stations, interpolating smooth functions to allow a differential formulation and then approximating this by a set of difference equations relative to a new grid seems somewhat inefficient. Several workers have suggested that the difference equations would be best if formulated and solved relative to the original network.
This doesn’t always work, however. Despite a double reformulation, the inefficient method often produces more sensible answers. This is not entirely fortuitous. We know that temperature, pressure and other meteorological variables are smooth functions of position, and so the differential formulation is, in principle, more correct. The meteorologists’ experience uses this fact, in the first reformulation, to supply data that, strictly speaking, isn’t there. The second reformulation is carried out to produce a grid that distorts the differential view as little as possible.[xviii]
In many cases, there are accepted procedures to be followed when a differential equation is approximated by a difference equation. The key to success is to choose a suitable ‘step-length’, that is a suitable temporal or spatial set of sampling points. If we choose a sufficiently small step-length h, then the exact solution will be approximated by the computed solution for (in the temporal case) a time T. Decreasing h increases T, the accuracy of the approximation and the labour and cost of the computation. If, to economise, we were to choose too large a value for h, the computed solution would soon become unacceptably different from the true one. This situation causes many difficulties in practical computation, although in principle it can always be avoided by use of a suitably small step-length. If the differential equation is asymptotically stable (and even under less stringent conditions), the error introduced when we replace the differential equation by a difference equation is small, and T is typically large.
If we begin with a difference equation, nature has, so to speak, chosen the step-length for us. We are given the initial value of some variable and a rule by which to calculate a ‘next value’. Application of the rule allows us to compute the value after one step-length; a further calculation gives us the value after two step-lengths, and so on. In principle, the system is perfectly determined once the rule and the initial value are given.
We might expect that for a single difference equation, or a system of two, the behaviour would settle down into a simple case of complete equilibrium or exact periodicity. Sometimes this does happen, but not always. There are, unfortunately, very simple systems which behave in quite complicated and apparently erratic ways.
What happens turns out to depend very much on the initial slate of the system. Some such initial values will give rise to equilibrium or periodic behaviour, but others (many more of them) lead to aperiodic and apparently random fluctuations. Moreover, asymptotic stability doesn’t apply, so that any error, no matter how small, in our initial value or in any subsequent value throws the entire calculation out of kilter.
The difficulty is that such errors are almost bound to occur. We must accept the possibility that our count of a population of cicadas (for example) could be out by one. Furthermore, as we calculate successive values of the variable (here population size), we face the necessity of rounding off the numbers involved in the calculations — that is to say, we necessarily introduce errors.[xix]
Furthermore, nature itself can thwart us here. Our rule for the calculation of successive values of the variable derives from the modelling process which omits as negligible all but a small number of factors. Once it is admitted that those excluded from the analysis could have any effect whatsoever, the whole calculation becomes useless, except for small values of T.
It is this situation that has been described as ‘chaotic’. The term is used in two senses. First, the typical solution of the difference equation shows no periodicity or ‘pattern’ whatsoever. Second, arbitrarily minute gaps in our knowledge render any attempt at forecasting beyond the immediate future impossible.
The cause of chaotic phenomena in difference equations may be understood with reference to the computer solution referred to earlier. If we take for definiteness a problem of population growth, we may plausibly suppose this to have associated with it some natural time-scale (e.g. the time taken for a population to double its size). The problem of choosing the correct step-length is essentially one of matching the imposed time-scale h to this natural time-scale. With differential equations, we may choose our value of h to achieve this. However, if a difference equation account is indicated, nature itself has specified both time-scales. Chaos ensues where these are ill-matched. This is precisely the problem with the weather-stations. The actual grid size given by the placement of the stations does not correspond necessarily to the natural scale of weather-patterns.
Chaos in difference equations seems to have been first studied systematically by the Hungarian mathematician Barna. The most thorough investigation, however, was an independent one by the Ukrainian, Sharkovsky. Most recently, and again independently, the U.S. workers Li and Yorke investigated the phenomenon.[xx]
Barna and Sharkovsky did not interest themselves in applications. The impact of chaos on science was first realised by the meteorologist and fluid dynamicist, E.N. Lorenz, who suggests that the weather obeys essentially chaotic dynamics and thus cannot be predicted with any accuracy for more than a few days ahead.[xxi]
More recently, the physicist turned biologist, R.M. May, has become interested in ecological applications.[xxii] There is some evidence to suggest that the severity of cicada plagues, for example, may be subject to the same uncertainty as next year’s weather.
Indeed, even with systems of differential equations, chaotic phenomena can occur. Such a system must, in view of the result quoted earlier, contain at least three equations. Even so, this is a relatively simple system. The first set of three differential equations to exhibit chaotic solutions was given by Lorenz.[xxiii]
Because the behaviour of the system is chaotic, Lorenz’s example is seen to involve a breakdown of asymptotic stability, for otherwise Sell’s theorem would be violated. This example throws considerable doubt on the methodology which assumes asymptotic stability to hold. We need to know a lot more about when we may expect it and when not.
Another disturbing example of an essentially chaotic phenomenon was recently given by Smale (like Thom, a Fields medallist).[xxiv] Smale considers a widely accepted system of equations describing competition between different species. He then proceeds to show that if there are more than four species involved, these equations are compatible with any observations whatsoever!
This last example raises one of the most disturbing implications of chaos. In modelling a real ecosystem (for example), we rely on some theoretical analysis or other to set up the basic equations of the mathematics. If two different researchers set up opposing systems, each of which predicted a chaotic solution, and if, furthermore, observation appeared to confirm the prediction, there would be no way at all to decide which theory accorded better with reality.
The same would be true of any other possible theory with chaotic dynamics. Quite what we make of this insight is not yet clear. One possibility is to dismiss as impossible large areas of mathematical modelling. Another, possibly more attractive, approach is to choose one’s theory on some ground additional to its (in this case, minimal) concordance with reality. Thus we might choose the basic equations for reasons of convenience or simplicity, the plausibility of the underlying assumptions or some such aesthetic grounds.
This would entail a rejection of the Popperian criterion of falsifiability as the mark of a good scientific theory, or at least a limitation of this principle. Interestingly enough, Thom has also argued against the Popperian view in justifying some of his applications of catastrophe theory.[xxv]
This is only one of a number of apparent, but deep, connections between catastrophe and chaos. This subject is still very largely unexplored. However, that some connection exists may be seen by considering a simple difference equation. If we allow adjustment of some aspect of this system (say the ratio of two time-scales) we reach a critical point beyond which the solutions are chaotic. Here we have a sudden change of behaviour, in other words a catastrophe.
In some cases, this will be one of Thom’s seven, but in other cases not. The reason for this is that not all such transitions satisfy the assumptions under which the theorem of the seven holds. Thom and others are currently attempting to produce more general counterparts to the theorem of the seven. Some interesting isolated results have emerged, but so far not the general theory envisaged. Whether it will ever be found remains to be seen. The full connection between catastrophe and chaos is unlikely to be known without the help of such a theory.
Catastrophe theory and the study of chaos are very recent branches of mathematics. They seem likely to be important, although we cannot yet say how important. We may be over-reacting to relatively minor advances in research; on the other hand, we may have our first glimpse of a radically altered world-view.
[i] The French catastrophe, as well as meaning ‘disaster’, may be translated as dénouement. It is a dramatic word in the original, and was purposely chosen to be so (Thom, interviewed for Times Higher Education Supplement, 5 December 1975), but it is not as startling as ‘catastrophe’ is in English. There was a brief attempt to coin an English equivalent, ‘catastrophy’, but this has been abandoned. Some mathematicians retain the French pronunciation or anglicise it to ‘cat-as-trough’; the normal English pronunciation, however, is more usual.
[ii] Stabilité Structurelle et Morphogénèse, by R. Thom, (Benjamin, N.Y., 1972). Now available from the same publisher in a translation by D.H. Fowler under the title Structural Stability and Morphogenesis (1974).
[iii] See, inter alia, Structural Stability and Morphogenesis, Chapter 13.
[iv] See, inter alia, his article ‘Catastrophe Theory’, Scientific American, April 1976.
[v] In G. Wassermann’s Stability of Unfoldings, (Springer, Berlin, 1974).
[vi] I. Stewart, ‘The Seven Elementary Catastrophes’, New Scientist, 20 November 1975.
[vii] E.C. Zeeman, op.cit. note 4.
[viii] C. Panati, ‘Catastrophe Theory’, Newsweek, 19 January 1976.
[ix] D. Dale, ‘A Theory that may Predict West Gate — and other Disasters’, National Times, 26-31 January 1976.
[x] For an extensive, but still incomplete, bibliography, see pp. 390-401 of Dynamical Systems — Warwick 1974, ed. A. Manning, (Springer, Berlin, 1975).
[xi] This point is expanded in Thom’s article ‘The Two-Fold Way of Catastrophe Theory’, appearing on pp. 235-252 of Structural Stability, the Theory of Catastrophes, and Applications in the Sciences, ed. P. Hilton, (Springer, Berlin, 1976).
[xii] J. Croll, ‘Is Catastrophe Theory Dangerous?’, New Scientist, 17 June 1976.
[xiii] For a fuller account, see my article, ‘Catastrophe Theory and Its Applications’, Mathematical Scientist, 1977.
[xiv] M. Deakin, ‘The Steady States of Ecosystems’, Mathematical Biosciences, 1975; Jane Cronin, ‘Periodic Solutions in n Dimensions and Volterra Equations’, Journal of Differential Equations, 1976.
[xv] G.R. Sell, ‘Periodic Solutions and Asymptotic Stability’, Journal of Differential Equations, 1966.
[xvi] See, for example, P. Erlich and C. Birch, ‘The “Balance of Nature” and “Population Control”’, American Naturalist, 1967.
[xvii] See, inter alia, R.M. May, ‘Deterministric Models with Chaotic Dynamics’, Nature, 17 July 1975.
[xviii] An excellent account is given by H.R. van der Vaart, ‘A Comparative Investigation of Certain Difference Equations and Related Differential Equations: Implications for Model-Building’, Bulletin of Mathematical Biology, 1973.
[xix] This introduction of error is quite clearly a necessity with a calculator or computer, which can hold only a limited number of digits. It occurs in practice, however, even if these are not used.
[xx] For references to the original literature, see P. Kloeden, M. Deakin and A. Tirkel, ‘A Precise Definition of Chaos’, Nature, 18 November 1976.
[xxi] E.N. Lorenz, ‘The Problem of Deducing the Climate from the Governing Equations’, Tellus, 1964.
[xxii] R.M. May, op. cit., note 17. May, who formerly held a chair in Physics at the University of Sydney, is now a Professor of Biology at Princeton.
[xxiii] E.N. Lorenz, ‘Deterministic Nonperiodic Flow’, Journal of the Atmospheric Sciences, 1963.
[xxiv] S. Smale, ‘On the Differential Equations of Species in Competition’, Journal of Mathematical Biology, 1976.
[xxv] The most explicit statement known to me occurs in a personal communication dated 8 April 1975. However, the attitude is implicit in much of Thom’s writing. See, for example, op.cil. note 11 or R. Thom, A. Lun and M. Deakin, Applications of Catastrophe Theory, Monash University, Mathematics Department, 1976.