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.
1
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,i 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.ii 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
2
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,iii 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 Anaxagorasiv 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,
3
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
4
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 (1880v) 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.
5
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
6
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,vi 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.vii
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 perioddoubling 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).
8
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),viii even though the discoverers did not understand what they had
discovered at the time.ix 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.x 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):
9
“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
10
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,xi 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
11
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
12
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
13
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.
14
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i
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.
ii
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.
iii
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.
iv
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).
v
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.
vi
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.
vii
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.
viii
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.
ix
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.
x
For a more complete discussion of such examples, see Rosser (1999).
xi
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.
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