CHAOS THEORY BEFORE LORENZ

 
 
 
 
 
 

CHAOS THEORY BEFORE LORENZ 

J. Barkley Rosser, Jr.

Department of Economics

James Madison University

Harrisonburg, VA 22807

rosserjb@jmu.edu 
 

September 2008 

Abstract: 

      We consider the precursors to the discovery of sensitive dependence on initial conditions by Edward Lorenz (1963) in his model of climatic fluid dynamics.  This will focus on work in various disciplines that imply either such sensitivity, irregular endogenous dynamic patterns, or fractal nature of an attractor, as is also found in the attractor underlying the model Lorenz studied.  Going from ancient hints in Anaxagoras through nineteenth century mathematics and physics, the main areas of such development will be argued to have been in celestial mechanics, oscillators, and economics. 
 
 
 

Acknowledgements:  The author wishes to thank William (“Buz”) Brock, the late Reid Bryson, Dick Day, Dee Dechert, Laura Gardini, Steve Guastello, Cars Hommes, Blake LeBaron, Hans-Walter Lorenz, Benoît Mandelbrot, Akio Matsumoto, Bruce Mizrach, Tönu Puu, Otto Rössler, and Don Saari for useful discussions of the issues in this paper over a long period of time.  
 
 
 
 
 
 
 
 
 
 
 

Introduction

      The late Edward Lorenz (1963) is justly famous for his discovery “on a coffee break” of sensitive dependence on initial conditions (SDIC), known popularly as the “butterfly effect,” the most widely agreed upon characteristic of chaotically dynamic nonlinear systems,¹ while studying computer simulations of a three equation model of climatic fluid dynamics.  While many think he used this term in his original 1963 paper, it does not appear there, having been coined by him for a 1972 presentation he made to a conference on climatology, although he himself has since speculated that the popularity of the phrase “butterfly effect” has arisen partly due to the vaguely butterfly appearance of the attractor bearing his name that is implied by the model he studied in his 1963 paper (Lorenz, 1993).  He has reported that he almost used a seagull instead (Lorenz, 1993, p. 15), noting that it was an old line among meteorologists that a man sneezing in China could cause people in New York to start shoveling snow (the original butterfly example being that a butterfly flapping its wings in Brazil could cause a tornado in Texas).

        While Lorenz’s discovery has achieved widespread attention that it did not initially receive upon its initial publication is well-deserved outcome, largely being read initially by meteorologists and climatologists who did not fully appreciate its broader mathematical implications.²  Nevertheless, the possibility of SDIC had been realized by others much earlier, including Maxwell (1876), Hadamard (1898), and Poincaré, implicitly in a model in 1890 and explicitly and consciously in 1908.  The idea of fractality also had a long history preceding him going back at least to Cantor (1883), with the idea of endogenously erratic dynamics that come close to following periodicity arguably going back as far as the pre-Socratic Greek philosopher, Anaxagoras, according to Rössler (1998), although erratic dynamics were first clearly shown by Cayley (1879). 

      This paper will discuss these earlier foundations and their development that preceded Lorenz’s important paper of 1963.  I shall not discuss much of the work that came after him in the 1960s and 1970s that would lead to the clearer understanding and codification of the concept,³ eventually resulting in a fad surrounding chaos theory after the publication of the popular bestseller by James Gleick (1987), who did much to publicize Lorenz’s role widely.  We shall look more at the development of SDIC, fractality, and irregularity of dynamics in such areas as pure mathematics, celestial mechanics, oscillators, and economics, keeping in mind that most of the people involved in these earlier developments did not understand fully the significance of what they were doing or how it would relate to this later development we now call chaos theory.  Indeed, many of these figures found the ideas they discovered or studied to be disturbing and even embarrassing, often relegating their discussion of them to footnotes or appendices where they did not clutter up the neater and simpler  results that they were focusing on in the main parts of their papers and books, Poincaré being an example of this attitude. 

The Earliest Predecessors

      One can argue that he is stretching in his interpretation, but Otto Rössler (1998, p. 3) argues strongly for the foundational role of Anaxagoras⁴ in his study of the unmixing by the mind of a deeply mixed reality, an emergence of the simple out of the complex.

      “Anaxagoras introduces a technical term…”around motion” (perichoresis)… He goes to great pains to make clear it is not a circular, a closed, a periodic motion which he has in mind…Anaxagoras single-handedly created the qualitative mathematical notions used so successfully later by the Poincaré school: deterministic flow, surface-de-section (cross-section through a flow, i.e. a recurrence) and – most important – the notions of mixing and unmixing.”

Rössler goes on to also argue that Anaxagoras had the idea of self-similarity that would later be associated with fractality, in that the roundabout motion starts in the small and goes to the large and that “the mind is self-similar (totally similar) both in the large and in the small” (Anaxagoras, 456 B.C.E., quoted on Rössler, 1998, p. 12).

      Yet another figure who has been seen to foreshadow modern chaos theory is the Renaissance polymath, Leonardo da Vinci.  In his case it is in some of his drawings in which he depicts wind.  These depict spectacular turbulence, and turbulence in fluid dynamics has long been a central area of study associated with chaos theory, most notably including the work of Lorenz himself.

      Another foreshadowing was in mathematics by Leibniz (1695) in connection with his independent invention of the calculus.  Unlike his rival, Newton, he posited the possibility of fractional derivatives.  These can be seen as predecessors to the later idea of non-integer dimensions that is central to fractal geometry.

      Finally, while these must be admitted to be rather vague, it has been argued that the philosophers Kant (Roqué, 1985) and Schelling (Heuser-Kessler, 1992) anticipated elements of the “dynamicist metaphysics” of Poincaré, whose own work is among the most important in the pre-Lorenz development of chaos theory. 

Late Nineteenth Century Developments in Mathematics

      The 1870s would be a time when developments in mathematics linked with thinking about physics would open doors on new ideas and approaches that would lead to the Lorenzian SDIC and also fractal geometry, as well as the possibility of endogenously erratic dynamics, even if these ideas were viewed by most as somewhat peculiar or even producing “monsters.”  One would be the first efforts to understand the basic idea of SDIC, how a small change in a dynamical system could lead to much larger changes in outcomes it could generate.  This would come with efforts by the figure who first developed the theoretical unification of electricity and magnetism, James Clerk Maxwell (1876). While he did not fully solve the problem, Louça (1997, p. 216) credits Maxwell with understanding the problem in examining “that class of phenomena that such that a spark kindles a forest, a rock creates an avalanche or a word prevents an action.”

      As reported by Mandelbrot (1983, p. 4), the decade would also see in 1872 the discovery by Weierstrass of a continuous but non-differentiable function that bears his name.  Such functions that are discontinuous in their first derivatives everywhere would be used by Lord Rayleigh (1880⁵) to study the frequency spectrum of blackbody radiation, with the lack of finite derivatives in certain bands suggesting the existence of infinite energy, which would come to be called the “ultraviolet catastrophe,” from which catastrophe theory would eventually obtain its moniker.  The resolution of this complication would involve the invention of quantum mechanics by Max Planck, as argued by Stewart (1989) and Ruelle (1991).  In any case, Mandelbrot saw the Weierstrass function as being a foundation for fractal geometry, many fractal sets exhibiting exactly this character of continuity but non-differentiability.

      The decade would close out with the suggestion by Sir Arthur Cayley (1879) to study iterations using Newton’s method of the simple cubic equation of the form

                         x³ – 1 = 0.                                                                   (1)

This equation possesses three roots, and Cayley inquired regarding which of the three the iterative process would converge to from an arbitrary point.  This proved to present considerable difficulties, with these iterations forming something like the fractal Julia set (1918) as argued by Peitgen, Jürgens, & Saupe (1992, pp. 774-775), which can also be seen as a form of erratic dynamics.  This also resembles somewhat the problem of the dynamics of a pendulum over three magnets, which generates fractal basin boundaries around the three different attractor sets.

      It would be in the following decade that Georg Cantor (1883) would discover the clearest example of a fractal set, the Cantor set (also known as Cantor dust or Cantor discontinuum).  Some dismissed this (and his discovery of transfinite sets) as “pathological,” a diagnosis enhanced by his stays in mental institutions, but his work is now regarded as fundamental for many branches of mathematics.  The basic Cantor set is constructed out of the closed [0, 1] interval by iteratively removing the open middle thirds from the interval and the subsequent remaining sub-intervals after each iteration.  What is left after an infinite set of iterations is infinitely subdividable, completely discontinuous, and nowhere dense, of measure zero length, while containing a continuum of points, and possessing a fractal dimension of ln2/ln3. 

      This monster would inspire a huge outpouring of imitators and followers over the next century as documented by Mandelbrot (1983) and Peitgen, Jürgens, & Saupe (1992), including the Lorenz attractor, with the first two in this line being the space-filling curves of Peano (1890) and Hilbert (1891).  Felix Hausdorff (1918) would formalize the definition of fractal dimension, and Richardson (1922), inspired by the Weierstrass function, would link this to the idea of a turbulent fluid dynamic consisting of a hierarchy of self-similar eddies, linked by a cascade without any overall measurable velocity, characterized by the following (Richardson, 1922, p. 6):

            “Big whorls have little whorls,

            Which feed on their velocity;

            And little whorls have lesser whorls,

            And so on to viscosity

            (in the molecular sense).”    

      All of these strands would come together in the culminating work in dynamic mathematics of the nineteenth century of Henri Poincaré.  Inspired substantially by the celestial mechanics question of the three body problem (Poincaré, 1890), he would develop the qualitative theory of differential equations, the concept of bifurcation that allows for the qualitative analysis of structural change of such systems,⁶ as well as show the possibility of strange attractors and SDIC (Poincaré, 1908), although Hadamard (1898) would beat him to the punch in showing SDIC explicitly in a model of flows on negatively curved geodesic surfaces.  More than any other figure, Poincaré   prefigured Lorenz with his dynamical systems that could exhibit the butterfly effect and erratically complex dynamics that would follow an “infinitely tight grid” like the fractal attractor that Lorenz’s system follows, even though Poincaré viewed these kinds of results he studied with a degree of disdain, if not outright horror.⁷ 

The Theory of Oscillators and the Discovery of Chaos

      Poincaré’s work inspired a stream of theoretical development in the theory of oscillators (Andronov, 1929) that would then be applied to problems of radio-engineering ((Mandel’shtam, Papaleski, Andronov, Vitt, Gorelik, & Khaikin, 1936), with this work influencing later work on turbulent dynamics (Kolmogorov, 1958).  While the Russians were carrying out this theoretical work, others were developing more specific models of oscillators that would later be shown to be capable of generating chaotic dynamics.  Not only were these models capable of doing so, but as we shall see, the first experimental discovery of a physically chaotic system arose out of these studies.

      The first of these was a model of an electro-magnetized vibrating beam due to Duffing (1918).  Early work that suggested the complex nature of dynamics that the Duffing oscillator model was capable of was carried out by Cartwright & Littlewood (1945), which inspired later study of its strange attractor by Ueda (1980).  This would be one of the first specific models that would be shown capable of generating period-doubling cascades of bifurcations in a transition to chaos as a crucial control parameter is varied (Holmes, 1979).

      More influential was the model of an electrical circuit with a triode valve whose resistance changes with the current due to van der Pol (1927).  The unforced version of this is given by

                  d²x/dt² + a(x² - b)x + x = 0,                                                      (2)

with a > 0 and b a control variable.  For b < 0 the origin is the attractor, but a b = 0 there is a bifurcation with a limit cycle occurring for b > 0, a Hopf bifurcation.  The forced version of this model is capable of generating fully chaotic dynamics (Levi, 1981).

      Which brings us to the experimental discovery of chaotic dynamics in 1927, the year van der Pol first wrote down his model, by van der Pol & van der Mark (1927).  Ironically, while they understood approximately what they had found, they did not fully understand the mathematics of it as this would not be fully established until 1981.  In any case, van der Pol & van der Mark were adjusting frequency ratios in telephone receivers and noticed zones where “an irregular noise is heard in the telephone receivers before the frequency jumps to the next lower value…[that]strongly reminds one of the tunes of a bagpipe” (van der Pol & van der Mark, 1927, p. 364). 

The Role of Economics in the Development of Chaos Theory

      Debate continues regarding whether or not true chaotic dynamics have been found in economic data or systems (Rosser, 2002, Chap. 7).  However, the study of economic models has contributed to the development of chaos theory, including the first case of the discovery of chaotic dynamics in a computer simulation model (Strotz, McAnulty, & Naines, 1953),⁸ even though the discoverers did not understand what they had discovered at the time.⁹  As with the theory of oscillators, models were developed that were only later discovered to be capable of generating chaotic dynamics.

      As also in other areas, there were invocations of possibly erratic dynamics in discussions of economic systems, prior to these being formulated in clear mathematical models.¹⁰  We shall note here two by famous early economists that appear to invoke some element of endogeneity as well as exogenous drivers.  The first is from the first Professor of Political Economy in Britain, Thomas Robert Malthus, in the first edition of his famous Essay on the Principle of Population  (1798, p. 33-34):

      “Such a history would tend greatly to elucidate the manner in which the constant check upon population acts, and would probably prove the existence of the retrograde and progressive movements that have been mentioned; though the times of their vibration must necessarily be rendered irregular, from the operation of many interrupting causes; such as, the introduction or failure of certain manufactures; a greater or lesser spirit of agricultural enterprise; years of plenty, or years of scarcity; wars and pestilence; poor laws; the invention of processes for shortening labour without the proportional extension of the market for the commodity; and particularly the difference between the nominal and the real price of labour; a circumstance, which has perhaps more than any other, contributed to conceal this oscillation from common view.”

This vision by Malthus would later trigger the development of specific demoeconomic models that were shown to be able to generate chaotic dynamics (Day, 1983).

      While Malthus emphasized the possibility of problems in the economy, often being viewed as a precursor of Keynes, Alfred Marshall is generally regarded as a main codifier of orthodox, neoclassical economic theory in Britain.  However, although no one has made a specific model based on these remarks, when Marshall would speak of the “real world,” he (and others as well), would sometimes invoke visions that did not fit so neatly with the generally orderly world that Marshall formulated in his economic theory, a world more consistent with nonlinear dynamics and even chaos.  Thus, from the final edition of his famous Principles of Economics (1920, p. 346) comes this:

      “But in reality such oscillations are seldom as rhythmical as those of a stone hanging freely from a string; the comparison would be more exact if the string were supposed to hang in the troubled waters of mill-race, whose stream was at one time allowed to flow freely, and at another partially cut off.  Nor are these complexities sufficient to illustrate the disturbances to which the economist and the merchant alike are forced to concern themselves.  If the person holding the string swings his hand with movements partly rhythmical, and partly arbitrary, the illustration will not outrun the difficulties of some very real and practical problems of value.”

It is worth noting that Marshall expanded this passage, adding the later complications in the later editions of this famous work, which first appeared in 1890, indicating his increasing awareness of such matters as his life experience accumulated.

        While there are a number of areas in economics where models would eventually show chaotic dynamics,¹¹ the one that led to the actual observation of chaotic dynamics in a computer simulation prior to Lorenz is that of macroeconomic dynamic modeling.  Inspired by the Great Depression, various models involving nonlinear relationships were developed from the mid-1930s to the early 1950s that could endogenously generate cyclical behavior (Rosser, 2000, Chap. 7).  Some of these were multiplier-accelerator models with nonlinear accelerator functions that drive investment (Hicks, 1950; Goodwin, 1951), while others simply had a more direct, nonlinear investment function (Kałecki, 1935; Kaldor, 1940).  While none of these economists were aware at the time of creating their models that they could generate such irregular dynamics, all of these models would eventually be shown capable of doing so.  In particular, it would be the Goodwin model that only two years after its publication would be shown able to do so by Strotz, McAnulty, & Naines (1953), even though neither they nor Goodwin understood properly what they had shown, although Goodwin would later study chaotic economic dynamics quite deeply (Goodwin, 1990).

      The original Goodwin model (1951) is given by three equations:

            c(t) = αy(t) – εdy(t)/dt + β(t),                                                             (3)

            dk(t)/dt = ρ[dy(t)/dt – θ)],                                                                   (4)

            y(t) = c(t) + dk(t)/dt + l(t),                                                                   (5)

with c(t) being consumption, y(t) income, k(t) the capital  stock (meaning that dk(t)/dt is induced investment), θ is a lag in the investment function, l(t) is autonomous investment, and β(t) is autonomous consumption.  This system can be compressed to

            εdy(t)/dt + (1-α)y(t) = ρ[dy(t-θ)/dy] + β(t) + l(t).                                 (6)

Goodwin showed that this could endogenously generate periodic cycles for certain not unreasonable values of the parameters.

      Ironically, given the work of van der Pol and van der Mark (1927), those who found the possibility of chaotic dynamics in this model did so by turning it into a model of electrical oscillation, an “electro-analog” model that they simulated on an analog computer (Strotz, McAnulty, & Naines, 1953).  Their equivalent model from electrical circuitry is

      RCdq(t)/dt + (1 – α)q(t) – ρ[da(t-θ)] = q₀(t),                                                   (7)

with q(t) representing electrical charge in coulombs, ρ[dq(t-θ)] a time-lagged function, R showing resistance in ohms, C as capacitance, with RC a constant..

      In simulating this model on an analog computer and testing it for various parameter values, they found it able to generate both a wide variety of cycles, an ability to jump from one cycle type to another, and to not follow any cycle at all, but to fluctuate erratically.  More significantly they realized the role of “initial conditions” in determining this wide variability of outcomes, recognizing that it could represent both differences in actual starting points as well as the role of exogenous shocks.  The final several paragraphs of their paper describe how these variabilities arise with many different

patterns possible with even small changes in the initial conditions, which they argued “actually enrich the explanatory value of the theory” [of Goodwin] (p. 408).  While they did not fully understand the significance of what they had found, they did understand that it was pretty interesting. 

Conclusions

      One might be tempted to infer that this recitation of all the predecessors to Edward Lorenz in previously discovering the various details of what he observed that day in 1961 when he took his coffee break demonstrates that he really did not do anything of any particularly great importance.  It was all already known, and Poincaré had even discovered all the parts, not just scattered ones as had other people, many of whom did not quite understand what they were observing..  However, this would be a misunderstanding.  Lorenz can be claimed to have “discovered chaos” both because he put all the elements together, sensitive dependence on initial conditions, with strange (fractal) attractors, and the resulting erratic dynamics, and because he saw them all together and in the context of a computer simulation situation that could be replicated.

      It is true that Poincaré had all the individual pieces.  It is also certainly true that these pieces had all been observed or contemplated even earlier by others.  It is also true that several individuals actually saw chaotic dynamics, either in the real world or in a computer simulation. But Poincaré never put them all together in a single model to point out their essential linkage.  Indeed, he shied away from his own findings.  And none of these others did so likewise, although some, such as Cantor, might well have not shied away in the manner of Poincaré seeking to understand an orderly universe.  It was Lorenz, who, once he realized that what he saw after his coffee break was not some fluke of the computer, focused on what it involved and went on to observe the beautiful strange attractor that underlay it and was so intimately associated with its sensitive dependence on initial conditions.

      Of course, it was not Lorenz who would derive the full mathematical understanding of what was going on.  But those who did so in the succeeding decades became aware of Lorenz’s work and his model, and it served to inspire their efforts in a way more forceful than all the earlier work that had been done, although of course these later thinkers would go back to Poincaré and ferret out that he had effectively done it all even earlier, even if he had not quite put it all together and emphasized what it was in the way that Lorenz would do with his model.

      Therefore, Edward Lorenz fully deserves the attention and praise he received for his efforts, which can be seen as a pivot point in the development of chaos theory, the moment when the scattered understandings and observations that had been floating around in the work of others over a long period of time became concentrated in a single model that revealed that there was a unified process and phenomenon at work.  In that regard, he did indeed “discover chaos” on that fateful day when he took his coffee break.   
 

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¹ There is considerable disagreement about the correct definition of mathematical chaos.  A widely used one is due to Devaney (1989): p. 50): that a map of a set into itself exhibit SDIC, that it is topologically transitive (indecomposable or irreducible), and that its periodic points are dense.  Mandelbrot (1983) has argued that fractality, or strangeness (non-integer dimension), of the attractor of a system be a crucial component (which Lorenz’s model exhibits), but most observers do not include this, thus allowing for “non-chaotic strange attractors.”  See Rosser (2000, Chap. 2) for further discussion.

² This author was first made aware of the idea of SDIC and chaos theory by the late climatologist, Reid Bryson, of the University of Wisconsin-Madison in 1973, who described Lorenz’s model and findings to me.  I note that at that time the term “chaos” had yet to be applied to this phenomenon, first appearing in print in the work of May (1974) and Li & Yorke (1975), with Yorke usually being credited for coining it for such dynamics.

³ Much work on mathematically clarifying the nature of strange attractors was done by Smale (1963), and Oseledec (1968) codified sufficiency conditions for the existence of SDIC.

⁴ Ironically, while “chaos” is originally a Greek word, Anaxagoras never used it in any of the extant fragments of his writing.  It is the original state of the universe in Greek mythology, and the word is used in English translations of Genesis from the Bible for the original state of the cosmos as well, although the original Semitic word was “Tohuwabohu” (Rössler, 1998, p 2).

⁵ Rayleigh (1916) would also develop the model that most clearly underlay the three equation model studied by Lorenz (1963).  Its ability to produce turbulent Bénard cells was what would attract Lorenz’s interest in it for studying the fluid dynamics of climatic and meteorological systems.

⁶ See Rosser (2000, Chap. 2) for further discussion.  Arnol’d (1992, Appendix) argues that the earliest precursor of bifurcation theory was Christiaan Huygens in 1654 with his study of the stability of cusp points in caustics and on wave fronts.

⁷ Thus, effectively Poincaré was the first to produce a model that could generate true mathematical chaos from his studies of the three body problem.  Stewart (1989, pp. 248-252) has suggested that the unpredictable “tumbling” of the rotation of Saturn’s moon Hyperion may well be an example of actual chaotic dynamics in celestial mechanics.

⁸ There may have been an earlier demonstration of chaotic dynamics in economics, although not through computer simulation, by Tord Palander (1935) in a regional economics model with a three-period cycle, although  as with others he did realize the possible implications of this.  It is not certain this is a chaotic model in that it is a more than one dimensional model, and the Li-Yorke (1975) theorem about three period cycles only applies to one dimensional models (it should be remembered that Li-Yorke is a special case of Sharkovsky, 1964).  An example of a two-dimensional system with a non-chaotic three period cycle would be a circle mapped into itself by rotating by a third each discrete peiod.

⁹ The first to present a model showing chaotic dynamics and suggesting that it could be applied to economics was Robert May (1976), with his logistic equation model.  The first to consciously present an economic model showing chaotic dynamics was David Rand (1978) of Cournot duopoly dynamics.

¹⁰ For a more complete discussion of such examples, see Rosser (1999).

¹¹ One such area involves cobweb models.  Some have argued that one of the early developers of such models (Ezekiel, 1938) understood that they could generate patterns of irregular dynamics, although he did not demonstrate this explicitly in his original paper, only making brief remarks suggesting it.