Chaos: Making a New Science - James Gleick (1988)
Notes on Sources and Further Reading
THIS BOOK DRAWS on the words of about two hundred scientists, in public lectures, in technical writing, and most of all in interviews conducted from April 1984 to December 1986. Some of the scientists were specialists in chaos; others were not. Some made themselves available for many hours over a period of months, offering insights into the history and practice of science that are impossible to credit fully. A few provided unpublished written recollections.
Few useful secondary sources of information on chaos exist, and the lay reader in search of further reading will find few places to turn. Perhaps the first general introduction to chaos—still eloquently conveying the flavor of the subject and outlining some of the fundamental mathematics—was Douglas R. Hofstadter’s November 1981 column in Scientific American, reprinted in Metamagical Themas (New York: Basic Books, 1985). Two useful collections of the most influential scientific papers are Hao Bai-Lin, Chaos (Singapore: World Scientific, 1984) and Predrag Cvitanović, Universality in Chaos (Bristol: Adam Hilger, 1984). Their selections overlap surprisingly little; the former is perhaps a bit more historically oriented. For anyone interested in the origins of fractal geometry, the indispensable, encyclopedic, exasperating source is Benoit Mandelbrot, The Fractal Geometry of Nature (New York: Freeman, 1977). The Beauty of Fractals, Heinz-Otto Peitgen and Peter H. Richter (Berlin: Springer-Verlag, 1986), delves into many areas of the mathematics of chaos in European-Romantic fashion, with invaluable essays by Mandelbrot, Adrien Douady, and Gert Eilenberger; it contains many lavish color and black-and–white graphics, several of which are reproduced in this book. A well-illustrated text directed at engineers and others seeking a practical survey of the mathematical ideas is H. Bruce Stewart and J. M. Thompson, Nonlinear Dynamics and Chaos (Chichester: Wiley, 1986). None of these books will be valuable to readers without some technical background.
In describing the events of this book and the motivations and perspectives of the scientists, I have avoided the language of science wherever possible, assuming that the technically aware will know when they are reading about integrability, power-law distribution, or complex analysis. Readers who want mathematical elaboration or specific references will find them in the chapter notes below. In selecting a few journal articles from the thousands that might have been cited, I chose either those which most directly influenced the events chronicled in this book or those which will be most broadly useful to readers seeking further context for ideas that interest them.
Descriptions of places are generally based on my visits to the sites. The following institutions made available their researchers, their libraries, and in some cases their computer facilities: Boston University, Cornell University, Courant Institute of Mathematics, European Centre for Medium Range Weather Forecasts, Georgia Institute of Technology, Harvard University, IBM Thomas J. Watson Research Center, Institute for Advanced Study, Lamont-Doherty Geophysical Observatory, Los Alamos National Laboratory, Massachusetts Institute of Technology, National Center for Atmospheric Research, National Institutes of Health, National Meteorological Center, New York University, Observatoire de Nice, Princeton University, University of California at Berkeley, University of California at Santa Cruz, University of Chicago, Woods Hole Oceanographic Institute, Xerox Palo Alto Research Center.
For particular quotations and ideas, the notes below indicate my principal sources. I give full citations for books and articles; where only a last name is cited, the reference is to one of the following scientists, who were especially helpful to my research:
William D. Bonner
Peter A. Carruthers
Richard J. Cohen
J. Doyne Farmer
Mitchell J. Feigenbaum
Ary L. Goldberger
Jerry P. Gollub
Ralph E. Gomory
Stephen Jay Gould
Douglas R. Hofstadter
John H. Hubbard
Raymond E. Ideker
Roderick V. Jensen
Thomas S. Kuhn
Cecil E. Leith
Edward N. Lorenz
Arnold J. Mandell
Paul C. Martin
Robert M. May
Francis C. Moon
Charles S. Peskin
Peter H. Richter
William M. Schaffer
Stephen H. Schneider
Michael F. Shlesinger
Yasha G. Sinai
Edward A. Spiegel
H. Bruce Stewart
William M. Visscher
Bruce J. West
Gareth P. Williams
Kenneth G. Wilson
Arthur T. Winfree
J. Austin Woods
James A. Yorke
LOS ALAMOS Feigenbaum, Carruthers, Campbell, Farmer, Visscher, Kerr, Hasslacher, Jen.
“I UNDERSTAND YOU’RE” Feigenbaum, Carruthers.
GOVERNMENT PROGRAM Buchal, Shlesinger, Wisniewski.
ELEMENTS OF MOTION Yorke.
PROCESS RATHER THAN STATE F. K. Browand, “The Structure of the Turbulent Mixing Layer,” Physica 18D (1986), p. 135.
THE BEHAVIOR OF CARS Japanese scientists took the traffic problem especially seriously; e.g., Toshimitsu Musha and Hideyo Higuchi, “The 1/f Fluctuation of a Traffic Current on an Expressway,” Japanese Journal of Applied Physics (1976), pp. 1271–75.
THAT REALIZATION Mandelbrot, Ramsey; Wisdom, Marcus; Alvin M. Saperstein, “Chaos—A Model for the Outbreak of War,” Nature 309 (1984), pp. 303–5.
“FIFTEEN YEARS AGO” Shlesinger.
JUST THREE THINGS Shlesinger.
THIRD GREAT REVOLUTION Ford.
“RELATIVITY ELIMINATED” Joseph Ford, “What Is Chaos, That We Should Be Mindful of It?” preprint, Georgia Institute of Technology, p. 12.
THE COSMOLOGIST John Boslough, Stephen Hawking’s Universe (Cambridge: Cambridge University Press, 1980); see also Robert Shaw, The Dripping Faucet as a Model Chaotic System (Santa Cruz: Aerial, 1984), p. 1.
THE BUTTERFLY EFFECT
THE SIMULATED WEATHER Lorenz, Malkus, Spiegel, Farmer. The essential Lorenz is a triptych of papers whose centerpiece is “Deterministic Nonperiodic Flow,” Journal of the Atmospheric Sciences 20 (1963), pp. 130–41; flanking this are “The Mechanics of Vacillation,” Journal of the Atmospheric Sciences 20 (1963), pp. 448–64, and “The Problem of Deducing the Climate from the Governing Equations,” Tellus 16 (1964), pp. 1–11. They form a deceptively elegant piece of work that continues to influence mathematicians and physicists twenty years later. Some of Lorenz’s personal recollections of his first computer model of the atmosphere appear in “On the Prevalence of Aperiodicity in Simple Systems,” in Global Analysis, eds. Mgrmela and J. Marsden (New York: Springer-Verlag, 1979), pp. 53–75.
THEY WERE NUMERICAL RULES A readable contemporary description by Lorenz of the problem of using equations to model the atmosphere is “Large-Scale Motions of the Atmosphere: Circulation,” in Advances in Earth Science, ed. P. M. Hurley (Cambridge, Mass.: The M.I.T. Press, 1966), pp. 95–109. An early, influential analysis of this problem is L. F. Richardson, Weather Prediction by Numerical Process (Cambridge: Cambridge University Press, 1922).
PURITY OF MATHEMATICS Lorenz. Also, an account of the conflicting pulls of mathematics and meteorology in his thinking is in “Irregularity: A Fundamental Property of the Atmosphere,” Crafoord Prize Lecture presented at the Royal Swedish Academy of Sciences, Stockholm, Sept. 28, 1983, in Tellus 36A (1984), pp. 98–110.
“IT WOULD EMBRACE” Pierre Simon de Laplace, A Philosophical Essay on Probabilities (New York: Dover, 1951).
“THE BASIC IDEA” Winfree.
“THAT’S THE KIND OF RULE” Lorenz.
SUDDENLY HE REALIZED “On the Prevalence,” p. 55.
SMALL ERRORS PROVED CATASTROPHIC Of all the classical physicists and mathematicians who thought about dynamical systems, the one who best understood the possibility of chaos was Jules Henri Poincaré. Poincaré remarked in Science and Method:
“A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still know the situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by the laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible….”
Poincaré’s warning at the turn of the century was virtually forgotten; in the United States, the only mathematician to seriously follow Poincaré’s lead in the twenties and thirties was George D. Birkhoff, who, as it happened, briefly taught a young Edward Lorenz at M.I.T.
THAT FIRST DAY Lorenz; also, “On the Prevalence,” p. 56.
“WE CERTAINLY HADN’T” Lorenz.
YEARS OF UNREAL OPTIMISM Woods, Schneider; a broad survey of expert opinion at the time was “Weather Scientists Optimistic That New Findings Are Near,” The New York Times, 9 September 1963, p. 1.
VON NEUMANN IMAGINED Dyson.
VAST AND EXPENSIVE BUREAUCRACY Bonner, Bengtsson, Woods, Leith.
FORECASTS OF ECONOMIC Peter B. Medawar, “Expectation and Prediction,” in Pluto’s Republic (Oxford: Oxford University Press, 1982), pp. 301–4.
THE BUTTERFLY EFFECT Lorenz originally used the image of a seagull; the more lasting name seems to have come from his paper, “Predictability; Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” address at the annual meeting of the American Association for the Advancement of Science in Washington, 29 December 1979.
SUPPOSE THE EARTH Yorke.
“PREDICTION, NOTHING” Lorenz, White.
THERE MUST BE A LINK “The Mechanics of Vacillation.”
FOR WANT OF A NAIL George Herbert; cited in this context by Norbert Wiener, “Nonlinear Prediction and Dynamics,” in Collected Works with Commentaries, ed. P. Masani (Cambridge, Mass.: The M.I.T. Press, 1981), 3:371. Wiener anticipated Lorenz in seeing at least the possibility of “self-amplitude of small details of the weather map.” He noted, “A tornado is a highly local phenomenon, and apparent trifles of no great extent may determine its exact track.”
“THE CHARACTER OF THE EQUATION” John von Neumann, “Recent Theories of Turbulence” (1949), in Collected Works, ed. A. H. Taub (Oxford: Pergamon Press, 1963), 6:437.
CUP OF HOT COFFEE “The predictability of hydrodynamic flow,” in Transactions of the New York Academy of Sciences II:25:4 (1963), pp. 409–32.
“WE MIGHT HAVE TROUBLE” Ibid., p. 410.
LORENZ TOOK A SET This set of seven equations to model convection was devised by Barry Saltzman of Yale University, whom Lorenz was visiting. Usually the Saltzman equations behaved periodically, but one version “refused to settle down,” as Lorenz said, and Lorenz realized that during this chaotic behavior four of the variables were approaching zero—thus they could be disregarded. Barry Saltzman, “Finite Amplitude Convection as an Initial Value Problem,” Journal of the Atmospheric Sciences 19 (1962), p. 329.
GEODYNAMO Malkus; the chaos view of the earth’s magnetic fields is still hotly debated, with some scientists looking for other, external explanations, such as blows from huge meteorites. An early exposition of the idea that the reversals come from chaos built into the system is K. A. Robbins, “A moment equation description of magnetic reversals in the earth,” Proceedings of the National Academy of Science 73 (1976), pp. 4297–4301.
WATER WHEEL Malkus.
THREE EQUATIONS This classic model, commonly called the Lorenz system, is:
dx/dt = 10(y-x)
dy/dt = – xz + 28x – y
dz/dt = xy–(8/3)z.
Since appearing in “Deterministic Nonperiodic Flow,” the system has been widely analyzed; one authoritative technical volume is Colin Sparrow, The Lorenz Equations, Bifurcations, Chaos, and Strange Attractors (Springer-Verlag, 1982).
“ED, WE KNOW” Malkus, Lorenz.
NO ONE THOUGHT “Deterministic Nonperiod Flow” was cited about once a year in the mid 1960s by the scientific community; two decades later, it was cited more than one hundred times a year.
THE HISTORIAN OF SCIENCE Kuhn’s understanding of scientific revolutions has been widely dissected and debated in the twenty-five years since he put it forward, at about the time Lorenz was programming his computer to model weather. For Kuhn’s views I have relied primarily on The Structure of Scientific Revolutions, 2nd ed. enl. (Chicago: University of Chicago Press, 1970) and secondarily on The Essential Tension: Selected Studies in Scientific Tradition and Change (Chicago: University of Chicago, 1977); “What Are Scientific Revolutions?” (Occasional Paper No. 18, Center for Cognitive Science, Massachusetts Institute of Technology); and Kuhn, interview. Another useful and important analysis of the subject is I. Bernard Cohen, Revolution in Science (Cambridge, Mass.: Belknap Press, 1985).
“I CAN’T MAKE Structure, pp. 62–65, citing J. S. Bruner and Leo Postman, “On the Perception of Incongruity: A Paradigm,” Journal of Personality XVIII (1949), p. 206.
MOPPING UP OPERATIONS structure, p. 24.
EXPERIMENTALISTS CARRY OUT Tension, p. 229.
IN BENJAMIN FRANKLIN’S STRUCTURE, pp. 13–15.
“UNDER NORMAL CONDITIONS TENSION, p. 234.
A PARTICLE PHYSICIST Cvitanović
TOLSTOY Ford, interview and “Chaos: Solving the Unsolvable, Predicting the Unpredictable,” in Chaotic Dynamics andFractals, ed. M. F. Barnsley and S. G. Demko (New York: Academic Press, 1985).
SUCH COINAGES But Michael Berry notes that the OED has “Chaology (rare) ‘the history or description of the chaos.’” Berry, “The Unpredictable Bouncing Rotator: A Chaology Tutorial Machine,” preprint, H. H. Wills Physics Laboratory, Bristol.
“IT’S MASOCHISM Richter.
THESE RESULTS APPEAR J. Crutchfield, M. Nauenberg and J. Rudnick, “Scaling for External Noise at the Onset of Chaos,” Physical Review Letters 46 (1981), p. 933.
THE HEART OF CHAOS Alan Wolf, “Simplicity and Universality in the Transition to Chaos,” Nature 305 (1983), p. 182.
CHAOS NOW PRESAGES Joseph Ford, “What is Chaos, That We Should Be Mindful of It?” preprint, Georgia Institute of Technology, Atlanta.
REVOLUTIONS DO NOT “What Are Scientific Revolutions?” p. 23.
“IT IS RATHER AS IF” Structure, p. 111.
THE LABORATORY MOUSE Yorke and others.
WHEN ARISTOTLE LOOKED “What Are Scientific Revolutions?” pp. 2–10.
“IF TWO FRIENDS” Galileo Opere VIII: 277. Also VIII: 129–30.
“PHYSIOLOGICAL AND PSYCHIATRIC” David Tritton, “Chaos in the swing of a pendulum,” New Scientist, 24 July 1986, p. 37. This is a readable, nontechnical essay on the philosophical implications of pendulum chaos.
THAT CAN HAPPEN In practice, someone pushing a swing can always produce more or less regular motion, presumably using an unconscious nonlinear feedback mechanism of his own.
YET, ODD AS IT SEEMS Among many analyses of the possible complications of a simple driven pendulum, a good summary is D. D’Humieres, M. R. Beasley, B. A. Huberman, and A. Libchaber, “Chaotic States and Routes to Chaos in the Forced Pendulum,” Physical Review A 26 (1982), pp. 3483–96.
SPACE BALLS Michael Berry researched the physics of this toy both theoretically and experimentally. In “The Unpredictable Bouncing Rotator” he describes a range of behaviors understandable only in the language of chaotic dynamics: “KAM tori, bifurcation of periodic orbits, Hamiltonian chaos, stable fixed points and strange attractors.”
FRENCH ASTRONOMER Hénon.
JAPANESE ELECTRICAL ENGINEER Ueda. 45 A YOUNG PHYSICIST Fox.
SMALE Smale, Yorke, Guckenheimer, Abraham. May, Feigenbaum; a brief, somewhat anecdotal account of Smale’s thinking during this period is “On How I Got Started in Dynamical Systems,” in Steve Smale, The Mathematics of Time: Essays on Dynamical Systems, Economic Processes, and Related Topics (New York: Springer-Verlag, 1980), pp. 147–51.
THE SCENE IN Moscow Raymond H. Anderson, “Moscow Silences a Critical American,” The New York Times, 27 August 1966, p. 1; Smale, “On the Steps of Moscow University,” The Mathematical Intelligencer6:2, pp. 21–27.
WHEN HE RETURNED Smale.
A LETTER FROM A COLLEAGUE The colleague was N. Levinson. Several threads of mathematics, running back to Poincaré, came together here. The work of Birkhoff was one. In England, Mary Lucy Cartwright and J. E. Littlewood pursued the hints turned up by Balthasar van der Pol in chaotic oscillators. These mathematicians were all aware of the possibility of chaos in simple systems, but Smale, like most well-educated mathematicians, was unaware of their work, until the letter from Levinson.
ROBUST AND STRANGE Smale; “On How I Got Started.”
IT WAS JUST A VACUUM TUBE van der Pol described his work in Nature 120 (1927), pp. 363–64.
“OFTEN AN IRREGULAR NOISE” Ibid.
TO MAKE A SIMPLE Smale’s definitive mathematical exposition of this work is “Differentiable Dynamical Systems,” Bulletin of the American Mathematical Society 1967, pp. 747–817 (also in The Mathematics of Time, pp. 1–82).
THE PROCESS MIMICS Rössler.
BUT FOLDING Yorke.
IT WAS A GOLDEN AGE Guckenheimer, Abraham.
“IT’S THE PARADIGM SHIFT Abraham.
A MODEST COSMIC MYSTERY Marcus, Ingersoll, Williams; Philip S. Marcus, “Coherent Vortical Features in a Turbulent Two-Dimensional Flow and the Great Red Spot of Jupiter,” paper presented at the 110th Meeting of the Acoustical Society of America, Nashville, Tennessee, 5 November 1985.
“THE RED SPOT ROARING” John Updike, “The Moons of Jupiter,” Facing Nature (New York: Knopf, 1985), p. 74.
VOYAGER HAD MADE Ingersoll; also, Andrew P. Ingersoll, “Order from Chaos: The Atmospheres of Jupiter and Saturn,” Planetary Report 4:3, pp. 8–11.
“YOU SEE THIS” Marcus.
“GEE, WHAT ABOUT” Marcus.
LIFE’S UPS AND DOWNS
RAVENOUS FISH May, Schaffer, Yorke, Guckenheimer. May’s famous review article on the lessons of chaos in population biology is “Simple Mathematical Models with Very Complicated Dynamics,” Nature 261 (1976), pp. 459–67. Also: “Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos,” Science 186 (1974), pp. 645–47, and May and George F. Oster, “Bifurcations and Dynamic Complexity in Simple Ecological Models,” The American Naturalist 110 (1976), pp. 573–99. An excellent survey of the development of mathematical modeling of populations, before chaos, is Sharon E. Kingsland, Modeling Nature: Episodes in the History of Population Ecology (Chicago: University of Chicago Press, 1985).
THE WORLD MAKES May and Jon Seger, “Ideas in Ecology: Yesterday and Tomorrow,” preprint, Princeton University, p. 25.
CARICATURES OF REALITY May and George F. Oster, “Bifurcations and Dynamic Complexity in Simple Ecological Models,” The American Naturalist 110 (1976), p. 573.
BY THE 1950s May.
REFERENCE BOOKS J. Maynard Smith, Mathematical Ideas in Biology (Cambridge: Cambridge University Press, 1968), p. 18; Harvey J. Gold, Mathematical Modeling of Biological Systems.
IN THE BACK May.
HE PRODUCED A REPORT Gonorrhea Transmission Dynamics and Control. Herbert W. Hethcote and James A. Yorke (Berlin: Springer-Verlag, 1984).
THE EVEN-ODD SYSTEM From computer simulations, Yorke found that the system forced drivers to make more trips to the filling station and to keep their tanks fuller all the time; thus the system increased the amount of gasoline sitting wastefully in the nation’s automobiles at any moment.
HE ANALYZED THE MONUMENT’S SHADOW Airport records later proved Yorke correct.
LORENZ’S PAPER Yorke.
“FACULTY MEMBERS” Murray Gell-Mann, “The Concept of the Institute,” in Emerging Syntheses in Science, proceedings of the founding workshops of the Santa Fe Institute (Santa Fe: The Santa Fe Institute, 1985), p. 11.
HE GAVE A COPY Yorke, Smale.
“IF YOU COULD WRITE” Yorke.
HOW NONLINEAR NATURE IS A readable essay on linearity, non-linearity, and the historical use of computers in understanding the difference is David Campbell, James P. Crutchfield, J. Doyne Farmer, and Erica Jen, “Experimental Mathematics: The Role of Computation in Nonlinear Science,” Communications of the Association for Computing Machinery 28 (1985), pp. 374–84.
“IT DOES NOT SAY” Fermi, quoted in S. M. Ulam, Adventures of a Mathematician (New York: Scribners, 1976). Ulam also describes the origin of another important thread in the understanding of non-linearity, the Fermi-Pasta–Ulam theorem. Looking for problems that could be computed on the new MANIAC computer at Los Alamos, the scientists tried a dynamical system that was simply a vibrating string—a simple model “having, in addition, a physically correct small non-linear term.” They found patterns coalescing into an unexpected periodicity. As Ulam recounts it: “The results were entirely different qualitatively from what even Fermi, with his great knowledge of wave motions, had expected…. To our surprise the string started playing a game of musical chairs, …” Fermi considered the results unimportant, and they were not widely published, but a few mathematicians and physicists followed them up, and they became a particular part of the local lore at Los Alamos. Adventures, pp. 226–28.
“NON ELEPHANT ANIMALS” quoted in “Experimental Mathematics,” p. 374.
“THE FIRST MESSAGE” Yorke.
YORKE’S PAPER Written with his student Tien-Yien Li. “Period Three Implies Chaos,” American Mathematical Monthly 82 (1975), pp. 985–92.
MAY CAME TO BIOLOGY May.
“WHAT THE CHRIST” May; it was this seemingly unanswerable question that drove him from analytic methods to numerical experimentation, meant to provide intuition, at least.
STARTLING THOUGH IT WAS Yorke.
A. N. SARKOVSKII “Coexistence of Cycles of a Continuous Map of a Line into Itself,” Ukrainian Mathematics Journal 16 (1964), p. 61.
SOVIET MATHEMATICIANS AND PHYSICISTS Sinai, personal communication, 8 December 1986.
SOME WESTERN CHAOS EXPERTS e.g., Feigenbaum, Cvitanović.
TO SEE DEEPER Hoppensteadt, May.
THE FEELING OF ASTONISHMENT Hoppensteadt.
WITHIN ECOLOGY May.
NEW YORK CITY MEASLES William M. Schaffer and Mark Kot, “Nearly One-dimensional Dynamics in an Epidemic,” Journal of Theoretical Biology 112 (1985), pp. 403–27; Schaffer, “Stretching and Folding in Lynx Fur Returns: Evidence for a Strange Attractor in Nature,” The American Naturalist 124 (1984), pp. 798–820.
THE WORLD WOULD BE “Simple Mathematical Models,” p. 467.
“THE MATHEMATICAL INTUITION” Ibid.
A GEOMETRY OF NATURE
A PICTURE OF REALITY Mandelbrot, Gomory, Voss, Barnsley, Richter, Mumford, Hubbard, Shlesinger. The Benoit Mandelbrot bible is The Fractal Geometry of Nature (New York: Freeman, 1977). An interview by Anthony Barcellos appears in Mathematical People, ed. Donald J. Albers and G. L. Alexanderson (Boston: Birkhäuser, 1985). Two essays by Mandelbrot that are less well known and extremely interesting are “On Fractal Geometry and a Few of the Mathematical Questions It Has Raised,” Proceedings of the Inter national Congress of Mathematicians, 16–14 August 1983, Warsaw, pp. 1661–75; and “Towards a Second Stage of Indeterminism in Science,” preprint, IBM Thomas J. Watson Research Center, Yorktown Heights, New York. Review articles on applications of fractals have grown too common to list, but two useful examples are Leonard M. Sander, “Fractal Growth Processes,” Nature 322 (1986), pp. 789–93; Richard Voss, “Random Fractal Forgeries: From Mountains to Music,” in Science and Uncertainty, ed. Sara Nash (London: IBM United Kingdom, 1985).
CHARTED ON THE OLDER MAN’S BLACKBOARD Houthakker, Mandelbrot.
WASSILY LEONTIEF Quoted in Fractal Geometry, p. 423.
INTRODUCED FOR A LECTURE Woods Hole Oceanographic Institute, August 1985.
BORN IN WARSAW Mandelbrot.
BOURBAKI Mandelbrot, Richter. Little has been written about Bourbaki even now; one playful introduction is Paul R. Halmos, “Nicholas Bourbaki,” Scientific American 196 (1957), pp. 88–89.
MATHEMATICS SHOULD BE SOMETHING Smale.
THE FIELD DEVELOPS Peitgen.
PIONEER-BY–NECESSITY “Second Stage,” p. 5.
THIS HIGHLY ABSTRACT Mandelbrot; Fractal Geometry, p. 74; J. M. Berger and Benoit Mandelbrot, “A New Model for the Clustering of Errors on Telephone Circuits,” IBM Journal of Research and Development 7 (1963), pp. 224–36.
THE JOSEPH EFFECT Fractal Geometry, p. 248.
CLOUDS ARE NOT SPHERES Ibid., p. 1, for example.
WONDERING ABOUT COASTLINES Ibid., p. 27.
THE PROCESS OF ABSTRACTION Ibid., p. 17.
“THE NOTION” Ibid., p. 18.
ONE WINTRY AFTERNOON Mandelbrot.
THE EIFFEL TOWER Fractal Geometry, p. 131, and “On Fractal Geometry,” p. 1663. 102 ORIGINATED BY MATHEMATICIANS F. Hausdorff and A. S. Besicovich.
“THERE WAS A LONG HIATUS” Mandelbrot.
IN THE NORTHEASTERN Scholz; C. H. Scholz and C. A. Aviles, “The Fractal Geometry of Faults and Faulting,” preprint, Lamont-Doherty Geophysical Observatory; C. H. Scholz, “Scaling Laws for Large Earthquakes,” Bulletin of the Seismological Society of America 72 (1982), pp. 1–14.
“A MANIFESTO” Fractal Geometry, p. 24.
“NOT A HOW-TO BOOK” Scholz.
“IT’S A SINGLE MODEL” Scholz.
“IN THE GRADUAL” William Bloom and Don W. Fawcett, A Textbook of Histology (Philadelphia: W. B. Saunders, 1975).
SOME THEORETICAL BIOLOGISTS One review of these ideas is Ary L. Goldberger, “Nonlinear Dynamics, Fractals, Cardiac Physiology, and Sudden Death,” in Temporal Disorder in Human Oscillatory Systems, ed. L. Rensing, U. An der Heiden, M. Mackey (New York: Springer-Verlag, 1987).
THE NETWORK OF SPECIAL FIBERS Goldberger, West.
SEVERAL CHAOS-MINDED CARDIOLOGISTS Ary L. Goldberger, Valmik Bhargava, Bruce J. West and Arnold J. Mandell, “On a Mechanism of Cardiac Electrical Stability: The Fractal Hypothesis,” Biophysics Journal48 (1985), p. 525.
WHEN E. I. DUPONT Barnaby J. Feder, “The Army May Have Matched the Goose,” The New York Times, 30 November 1986, 4:16.
“I STARTED LOOKING” Mandelbrot.
HIS NAME APPEARED I. Bernard Cohen, Revolution in Science (Cambridge, Mass.: Belknap, 1985), p. 46.
“OF COURSE, HE IS A BIT” Mumford.
“HE HAD SO MANY DIFFICULTIES” Richter.
IF THEY WANTED TO AVOID Just as Mandelbrot later could avoid the credit routinely given to Mitchell Feigenbaum in references to Feigenbaum numbers and Feigenbaum universality. Instead, Mandelbrot habitually referred to P. J. Myrberg, a mathematician who had studied iterates of quadratic mappings in the early 1960s, obscurely.
“MANDELBROT DIDN’T HAVE EVERYBODY’S” Richter.
“THE POLITICS AFFECTED” Mandelbrot.
EXXON’S HUGE RESEARCH FACILITY Klafter.
ONE MATHEMATICIAN TOLD FRIENDS Related by Huberman.
“WHY IS IT THAT” “Freedom, Science, and Aesthetics,” in Schönheit im Chaos, p. 35.
“THE PERIOD HAD NO SYMPATHY” John Fowles, A Maggot (Boston: Little, Brown, 1985), p. 11.
“WE HAVE THE ASTRONOMERS” Robert H. G. Helleman, “Self-Generated Behavior in Nonlinear Mechanics,” in Fundamental Problems in Statistical Mechanics 5, ed. E. G. D. Cohen (Amsterdam: North-Holland, 1980), p. 165.
BUT PHYSICISTS WANTED MORE Leo Kadanoff, for example, asked “Where is the physics of fractals?” in Physics Today, February 1986, p. 6, and then answered the question with a new “multi-fractal” approach in Physics Today, April 1986, p. 17, provoking a typically annoyed response from Mandelbrot, Physics Today, September 1986, p. 11. Kadanoff’s theory, Mandelbrot wrote, “fills me with the pride of a father—soon to be a grandfather?”
THE GREAT PHYSICISTS Ruelle, Hénon, Rössler, Sinai, Feigenbaum, Mandelbrot, Ford, Kraichnan. Many perspectives exist on the historical context for the strange-attractor view of turbulence. A worthwhile introduction is John Miles, “Strange Attractors in Fluid Dynamics,” in Advances in Applied Mechanics 24 (1984), pp. 189, 214. Ruelle’s most accessible review article is “Strange Attractors,” Mathematical Intelligencer 2 (1980), pp. 126–37; his catalyzing proposal was David Ruelle and Floris Takens, “On the Nature of Turbulence,” Communications in Mathematical Physics 20 (1971), pp. 167–92; his other essential papers include “Turbulent Dynamical Systems,” Proceedings of the International Congress of Mathematicians, 16–24 August 1983, Warsaw, pp. 271–86; “Five Turbulent Problems,” Physica 7D (1983), pp. 40–44; and “The Lorenz Attractor and the Problem of Turbulence,” in Lecture Notes in Mathematics No. 565 (Berlin: Springer-Verlag, 1976), pp. 146–58.
THERE WAS A STORY Many versions of this exist. Orszag cites four substitutes for Heisenberg—von Neumann, Lamb, Sommerfeld, and von Karman—and adds, “I imagine if God actually gave an answer to these four people it would be different in each case.”
THIS ASSUMPTION Ruelle; also “Turbulent Dynamical Systems,” p. 281.
TEXT ON FLUID DYNAMICS L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Oxford: Pergamon, 1959).
THE OSCILLATORY, THE SKEWED VARICOSE Malkus.
“THAT’S TRUE” Swinney.
IN 1973 SWINNEY Swinney, Gollub.
“IT WAS A STRING-AND–SEALING-WAX” Dyson.
“SO WE READ THAT” Swinney.
WHEN THEY BEGAN REPORTING Swinney, Gollub.
“THERE WAS THE TRANSITION” Swinney.
EXPERIMENT FAILED TO CONFIRM J. P. Gollub and H. L. Swinney, “Onset of Turbulence in a Rotating Fluid,” Physical Review Letters 35 (1975), p. 927. These first experiments only opened the door to an appreciation of the complex spatial behaviors that could be produced by varying the few parameters of flow between rotating cylinders. The next few years identified patterns from “corkscrew wavelets” to “wavy inflow and outflow” to “interpenetrating spirals.” A summary is C. David Andereck, S. S. Liu, and Harry L. Swinney, “Flow Regimes in a Circular Couette System with Independently Rotating Cylinders,” Journal of Fluid Mechanics 164 (1986), pp. 155–83.
DAVID RUELLE SOMETIMES SAID Ruelle. 132.
“ALWAYS NONSPECIALISTS FIND” Ruelle.
HE WROTE A PAPER “On the Nature of Turbulence.”
OPINIONS STILL VARIED They quickly discovered that some of their ideas had already appeared in the Russian literature; “on the other hand, the mathematical interpretation which we give of turbulence seems to remain our own responsibility!” they wrote. “Note Concerning Our Paper ‘On the Nature of Turbulence,’” Communications in Mathematical Physics 23 (1971), pp. 343–44.
PSYCHOANALYTICALLY “SUGGESTIVE” Ruelle.
“DID YOU EVER ASK GOD” “Strange Attractors,” p. 131.
“TAKENS HAPPENED” Ruelle.
“SOME MATHEMATICIANS IN CALIFORNIA” Ralph H. Abraham and Christopher D. Shaw, Dynamics: The Geometry of Behavior (Santa Cruz: Aerial: 1984).
“IT ALWAYS BOTHERS ME” Richard P. Feynman, The Character of Physical Law (Cambridge, Mass.: The M.I.T. Press, 1967), p. 57.
DAVID RUELLE SUSPECTED Ruelle.
THE REACTION OF THE SCIENTIFIC PUBLIC “Turbulent Dynamical Systems,” p. 275.
EDWARD LORENZ HAD ATTACHED “Deterministic Nonperiodic Flow,” p. 137.
“IT IS DIFFICULT TO RECONCILE Ibid., p. 140.
HE WENT TO VISIT LORENZ Ruelle.
“DON’T FORM A SELFISH CONCEPT Ueda reviews his early discoveries from the point of view of electrical circuits in “Random Phenomena Resulting from Nonlinearity in the System Described by Duffing’s Equation,” in International Journal of Non-Linear Mechanics 20 (1985), pp. 481–91, and gives a personal account of his motivation and the cool response of his colleagues in a postscript. Also, Stewart, private communication.
“A SAUSAGE IN A SAUSAGE” Rössler.
THE MOST ILLUMINATING STRANGE ATTRACTOR Hénon; he reported his invention in “A Two-Dimensional Mapping with a Strange Attractor,” in Communications in Mathematical Physics 50 (1976), pp. 69–77, and Michel Hénon and Yves Pomeau, “Two Strange Attractors with a Simple Structure,” in Turbulence and the Navier-Stokes Equations, ed. R. Teman (New York: Springer-Verlag, 1977).
IS THE SOLAR SYSTEM Wisdom.
“TO HAVE MORE FREEDOM” Michel Hénon and Carl Heiles, “The Applicability of the Third Integral of Motion: Some Numerical Experiments,” Astronomical Journal 69 (1964), p. 73.
AT THE OBSERVATORY Hénon.
“I, TOO, WAS CONVINCED” Hénon.
“HERE COMES THE SURPRISE” “The Applicability,” p. 76.
“BUT THE MATHEMATICAL APPROACH” Ibid., p. 79.
A VISITING PHYSICIST Yves Pomeau.
“SOMETIMES ASTRONOMERS ARE FEARFUL” Hénon.
OTHERS ASSEMBLED MILLIONS Ramsey.
“I HAVE NOT SPOKEN” “Strange Attractors,” p. 137.
“YOU CAN FOCUS” Feigenbaum. Feigenbaum’s crucial papers on universality are “Quantitative Unversality for a Class of Nonlinear Transformations,” Journal of Statistical Physics 19 (1978), pp. 25–52, and “The Universal Metric Properties of Nonlinear Transformations,” Journal of Statistical Physics 21 (1979), pp. 669–706; a somewhat more accessible presentation, though still requiring some mathematics, is his review article, “Universal Behavior in Nonlinear Systems,” Los Alamos Science 1 (Summer 1981), pp. 4–27. I also relied on his unpublished recollections, “The Discovery of Universality in Period Doubling.”
WHEN FEIGENBAUM CAME TO LOS ALAMOS Feigenbaum, Carruthers, Cvitanović, Campbell, Farmer, Visscher, Kerr, Hasslacher, Jen.
“IF YOU HAD SET UP” Carruthers.
THE MYSTERY OF THE UNIVERSE Feigenbaum.
OCCASIONALLY AN ADVISOR Carruthers.
AS KADANOFF VIEWED Kadanoff.
“THE CEASELESS MOTION” Gustav Mahler, letter to Max Marschalk.
“WITH LIGHT POISE” Goethe’s Zür Farbenlehre is now available in several editions. I relied on the beautifully illustrated Goethe’s Color Theory, ed. Rupprecht Matthaei, trans. Herb Aach (New York: Van Nostrand Reinhold, 1970); more readily available is Theory of Colors (Cambridge, Mass.: The M.I.T. Press, 1970), with an excellent introduction by Deane B. Judd.
THIS ONE INNOCENT-LOOKING EQUATION At one point, Ulam and von Neumann used its chaotic properties as a solution to the problem of generating random numbers with a finite digital computer.
TO METROPOLIS, STEIN, AND STEIN This paper—the sole pathway from Stanislaw Ulam and John von Neumann to James Yorke and Mitchell Feigenbaum—is “On Finite Limit Sets for Transformations on the Unit Interval,” Journal of Combinatorial Theory 15 (1973), pp. 25–44.
DOES A CLIMATE EXIST “The Problem of Deducing the Climate from the Governing Equations,” Tellus 16 (1964), pp. 1–11.
THE WHITE EARTH CLIMATE Manabe.
HE KNEW NOTHING OF LORENZ Feigenbaum.
THE SAME COMBINATIONS OF R’S AND L’S “On Finite Limit Sets,” pp. 30–31. The crucial hint: “The fact that these patterns … are a common property of four apparently unrelated transformations … suggests that the pattern sequence is a general property of a wide class of mappings. For this reason we have called this sequence of patterns the U-sequence where ‘U’ stands (with some exaggeration) for ‘universal.’” But the mathematicians never imagined that the universality would extend to actual numbers; they made a table of 84 different parameter values, each taken to seven decimal places, without observing the geometrical relationships hidden there.
“THE WHOLE TRADITION OF PHYSICS” Feigenbaum.
HIS FRIENDS SPECULATED Cvitanović.
SUDDENLY YOU COULD SEE Ford.
PRIZES AND AWARDS The MacArthur fellowship; the 1986 Wolf Prize in physics.
“IT WAS A VERY HAPPY” Gilmore.
BUT ALL THE WHILE Cvitanović.
WORK BY OSCAR E. LANFORD Even then, the proof was unorthodox in that it depended on tremendous amounts of numerical calculation, so that it could not be carried out or checked without the use of a computer. Lanford; Oscar E. Lanford, “A Computer-Assisted Proof of the Feigenbaum Conjectures,” Bulletin of the American Mathematical Society 6 (1982), p. 427; also, P. Collet, J.P. Eckmann, and O. E. Lanford, “Universal Properties of Maps on an Interval,” Communications in Mathematical Physics 81 (1980), p. 211.
“SIR, DO YOU MEAN” Feigenbaum; ”The Discovery of Universality,” p. 17.
IN THE SUMMER OF 1977 Ford, Feigenbaum, Lebowitz.
“MITCH HAD SEEN UNIVERSALITY” Ford.
“SOMETHING DRAMATIC HAPPENED” Feigenbaum.
“ALBERT IS GETTING MATURE” Libchaber, Kadanoff.
HE SURVIVED THE WAR Libchaber.
“HEUUM IN A SMALL BOX” Albert Libchaber, “Experimental Study of Hydrodynamic Instabilities. Rayleigh-Benard Experiment: Helium in a Small Box,” in Nonlinear Phenomena at Phase Transitions and Instabilities, ed. T. Riste (New York: Plenum, 1982), p. 259.
THE LABORATORY OCCUPIED Libchaber, Feigenbaum.
“SCIENCE WAS CONSTRUCTED” Libchaber.
“BUT YOU KNOW THEY DO!” Libchaber.
“THE FLECKED RIVER” Wallace Stevens, “This Solitude of Cataracts,” The Palm at the End of the Mind, ed. Holly Stevens (New York: Vintage, 1972), p. 321.
“INSOLID BILLOWING OF THE SOLID” “Reality Is an Activity of the Most August Imagination,” Ibid., p. 396.
“BUILDS ITS OWN BANKS” Theodor Schwenk, Sensitive Chaos (New York: Schocken, 1976), p. 19.
“ARCHETYPAL PRINCIPLE” Ibid.
“THIS PICTURE OF STRANDS” Ibid., p. 16.
“THE INEQUALITIES” Ibid., p. 39.
“IT MAY BE” D’Arcy Wentworth Thompson, On Growth and Form, J. T. Bonner, ed. (Cambridge: Cambridge University Press, 1961), p. 8.
“BEYOND COMPARISON THE FINEST” Ibid., p. viii.
“FEW HAD ASKED” Stephen Jay Gould, Hen’s Teeth and Horse’s Toes (New York: Norton, 1983), p. 369.
“DEEP-SEATED RHYTHMS OF GROWTH” On Growth and Form, p. 267.
“THE INTERPRETATION IN TERMS OF FORCE” Ibid., p. 114.
IT WAS SO SENSITIVE Campbell.
“IT WAS CLASSICAL PHYSICS” Libchaber.
NOW, HOWEVER, A NEW FREQUENCY Libchaber and Maurer, 1980 and 1981. Also Cvitanović’s introduction gives a lucid summary.
“THE NOTION THAT THE ACTUAL” Hohenberg.
“THEY STOOD AMID THE SCATTERED” Feigenbaum, Libchaber.
“YOU HAVE TO REGARD IT” Gollub.
A VAST BESTIARY OF LABORATORY EXPERIMENTS The literature is equally vast. One summary of the early melding of theory and experiment in a variety of systems is Harry L. Swinney, “Observations of Order and Chaos in Nonlinear Systems,” Physica 7D (1983), pp. 3–15; Swinney provides a list of references divided into categories, from electronic and chemical oscillators to more esoteric kinds of experiments.
TO MANY, EVEN MORE CONVINCING Valter Franceschini and Claudio Tebaldi, “Sequences of Infinite Bifurcations and Turbulence in a Five-Mode Truncation of the Navier-Stokes Equations,” Journal of Statistical Physics 21 (1979), pp. 707–26.
IN 1980 A EUROPEAN GROUP P. Collet, J.–P. Eckmann, and H. Koch, “Period Doubling Bifurcations for Families of Maps on Rn,” Journal of Statistical Physics 25 (1981), p. 1.
“A PHYSICIST WOULD ASK ME” Libchaber.
IMAGES OF CHAOS
MICHAEL BARNSLEY MET Barnsley.
RUELLE SHUNTED IT BACK Barnsley.
JOHN HUBBARD, AN AMERICAN Hubbard; also Adrien Douady, “Julia Sets and the Mandelbrot Set,” in pp. 161–73. The main text of The Beauty of Fractals also give a mathematical summary of Newton’s method, as well as the other meeting grounds of complex dynamics discussed in this chapter.
“NOW, FOR EQUATIONS” “Julia Sets and the Mandelbrot Set,” p. 170.
HE STILL PRESUMED Hubbard.
A BOUNDARY BETWEEN TWO COLORS Hubbard; The Beauty of Fractals; Peter H. Richter and Heinz-Otto Peitgen, “Morphology of Complex Boundaries,” Bunsen-Gesellschaft für Physikalische Chemie 89 1985), pp. 575–88.
THE MANDELBROT SET A readable introduction, with instructions for writing a do-it–yourself microcomputer program, is A. K. Dewdney, “Computer Recreations,” Scientific American (August 1985), pp. 16–32. Peitgen and Richter in The Beauty of Fractals offer a detailed review of the mathematics, as well as some of the most spectacular pictures available.
THE MOST COMPLEX OBJECT Hubbard, for example.
“YOU OBTAIN AN INCREDIBLE VARIETY “Julia Sets and the Mandelbrot Set,” p. 161.
IN 1979 MANDELBROT DISCOVERED Mandelbrot, Laff, Hubbard. A first-person account by Mandelbrot is “Fractals and the Rebirth of Iteration Theory,” in The Beauty of Fractals, pp. 151–60.
AS HE TRIED CALCULATING Mandelbrot; The Beauty of Fractals.
MANDELBROT STARTED WORRYING Mandelbrot.
NO TWO PIECES ARE “TOGETHER” Hubbard.
“EVERYTHING WAS VERY GEOMETRIC” Peitgen.
AT CORNELL, MEANWHILE Hubbard.
RICHTER HAD COME TO COMPLEX SYSTEMS Richter.
“IN A BRAND NEW AREA” Peitgen.
“RIGOR IS THE STRENGTH” Peitgen.
FRACTAL BASIN BOUNDARIES Yorke; a good introduction, for the technically inclined, is Steven W. MacDonald, Celso Grebogi, Edward Ott, and James A. Yorke, “Fractal Basin Boundaries,” Physica 17D (1985), pp. 125–83.
AN IMAGINARY PINBALL MACHINE Yorke.
“NOBODY CAN SAY” Yorke, remarks at Conference on Perspectives in Biological Dynamics and Theoretical Medicine, National Institutes of Health, Bethesda, Maryland, 10 April 1986.
TYPICALLY, MORE THAN THREE-QUARTERS Yorke.
THE BORDER BETWEEN CALM AND CATASTROPHE Similarly, in a text meant to introduce chaos to engineers, H. Bruce Stewart and J. M. Thompson warned: “Lulled into a false sense of security by his familiarity with the unique response of a linear system, the busy analyst or experimentalist shouts ‘Eureka, this is the solution,’ once a simulation settles onto an equilibrium of steady cycle, without bothering to explore patiently the outcome from different starting conditions. To avoid potentially dangerous errors and disasters, industrial designers must be prepared to devote a greater percentage of their effort into exploring the full range of dynamic responses of their systems.” Nonlinear Dynamics and Chaos (Chichester; Wiley, 1986), p. xiii.
“PERHAPS WE SHOULD BELIEVE” The Beauty of Fractals, p. 136.
WHEN HE WROTE ABOUT e.g., “Iterated Function Systems and the Global Construction of Fractals,” Proceedings of the Royal Society of London A 399 (1985), pp. 243–75.
“IF THE IMAGE IS COMPLICATED” Barnsley.
“THERE IS NO RANDOMNESS” Hubbard.
“RANDOMNESS IS A RED” Barnsley.
THE DYNAMICAL SYSTEMS COLLECTIVE
SANTA CRUZ Farmer, Shaw, Crutchfield, Packard, Burke, Nauenberg, Abrahams, Guckenheimer. The essential Robert Shaw, applying information theory to chaos, is The Dripping Faucet as a Model Chaotic System (Santa Cruz: Aerial, 1984), along with “Strange Attractors, Chaotic Behavior, and Information Theory,” Zeitschrift für Naturforschung 36a (1981), p. 80. An account of the roulette adventures of some of the Santa Cruz students, conveying much of the color of these years, is Thomas Bass, The Eudemonic Pie (Boston: Houghton Mifflin, 1985).
HE DID NOT KNOW Shaw.
WILLIAM BURKE, a SANTA CRUZ COSMOLOGIST Burke, Spiegel.
“COSMIC ARRHYTHMIAS” Edward A. Spiegel, “Cosmic Arrhythmias,” in Chaos in Astrophysics, J. R. Buchler et al., eds. (New York: D. Reidel, 1985), pp. 91–135.
THE ORIGINAL PLANS Farmer, Crutchfield.
BY BUILDING UP Shaw, Crutchfield, Burke.
A FEW MINUTES LATER Shaw.
“ALL YOU HAVE TO DO” Abraham.
DOYNE FARMER Farmer is the main figure and Packard is a secondary figure in The Eudemonic Pie, the story of the roulette project, written by a sometime associate of the group.
PHYSICS AT SANTA CRUZ Burke, Farmer, Crutchfield.
FORD HAD ALREADY DECIDED Ford.
THEY REALIZED THAT MANY SORTS Shaw, Farmer.
INFORMATION THEORY The classic text, still quite readable, is Claude E. Shannon and Warren Weaver, The Mathematical Theory of Communication (Urbana: University of Illinois, 1963), with a helpful introduction by Weaver.
“WHEN ONE MEETS THE CONCEPT” Ibid., p. 13.
NORMAN PACKARD WAS READING Packard.
IN DECEMBER 1977 Shaw.
WHEN LORENZ WALKED INTO THE ROOM Shaw, Farmer.
HE FINALLY MAILED HIS PAPER “Strange Attractors, Chaotic Behavior, and Information Flow.”
A. N. KOLMOGOROV AND YASHA SINAI Sinai, private communication.
AT THE PINNACLE Packard.
“YOU DON’T SEE SOMETHING” Shaw.
“IT’S A SIMPLE EXAMPLE” Shaw.
SYSTEMS THAT THE SANTA CRUZ GROUP Farmer; a dynamical systems approach to the immune system, modeling the human body’s ability to “remember” and to recognize patterns creatively, is outlined in J. Doyne Farmer, Norman H. Packard, and Alan S. Perelson, “The Immune System, Adaptation, and Machine Learning,” preprint, Los Alamos National Laboratory, 1986.
ONE IMPORTANT VARIABLE The Dripping Faucet, p. 4.
“A STATE-OF–THE-ART COMPUTER CALCULATION” Ibid.
A “PSEUDOCOLLOQUIUM” Crutchfield.
“IT TURNS OUT” Shaw.
“WHEN YOU THINK ABOUT A VARIABLE” Farmer.
RECONSTRUCTING THE PHASE SPACE These methods, which became a mainstay of experimental technique in many different fields, were greatly refined and extended by the Santa Cruz researchers and other experimentalists and theorists. One of the key Santa Cruz proposals was Norman H. Packard, James P. Crutchfield, J. Doyne Farmer, and Robert S. Shaw [the canonical byline list], “Geometry from a Time Series,” Physical Review Letters 47 (1980), p. 712. The most influential paper on the subject by Floris Takens was “Detecting Strange Attractors in Turbulence,” in Lecture Notes in Mathematics 898, D. A. Rand and L. S. Young, eds. (Berlin: Springer-Verlag, 1981), p. 336. An early but fairly broad review of the techniques of reconstructing phase-space portraits is Harold Froehling, James P. Crutchfield, J. Doyne Farmer, Norman H. Packard, and Robert S. Shaw, “On Determining the Dimension of Chaotic Flows,” Physica 3D (1981), pp. 605–17.
“GOD, WE’RE STILL” Crutchfield.
SOME PROFESSORS DENIED e.g., Nauenberg.
“WE HAD NO ADVISOR” Shaw.
MORE INTERESTED IN REAL SYSTEMS Not that the students ignored maps altogether. Crutchfield, inspired by May’s work, spent so much time in 1978 making bifurcation diagrams that he was barred from the computer center’s plotter. Too many pens had been destroyed laying down the thousands of dots.
LANFORD LISTENED POLITELY Farmer.
“IT WAS MY NAIVETÉ” Farmer.
“AUDIOVISUAL AIDS” Shaw.
ONE DAY BERNARDO HUBERMAN crutchfield, huberman.
“IT WAS ALL VERY VAGUE” Huberman.
THE FIRST PAPER Bernardo A. Huberman and James P. Crutchfield, “Chaotic States of Anharmonic Systems in Periodic Fields,” Physical Review Letters 43 (1979), p. 1743.
FARMER WAS ANGERED Crutchfield.
CLIMATE SPECIALISTS This is a continuing debate in the journal Nature, for example.
ECONOMISTS ANALYZING STOCK MARKET Ramsey.
FRACTAL DIMENSION, HAUSDORFF DIMENSION J. Doyne Farmer, Edward Ott, and James A. Yorke, “The Dimension of Chaotic Attractors,” Physica 7D (1983), pp. 153–80.
“THE FIRST LEVEL OF KNOWLEDGE” Ibid., p. 154.
HUBERMAN LOOKED OUT Huberman, Mandell (interviews and remarks at Conference on Perspectives in Biological Dynamics and Theoretical Medicine, Bethesda, Maryland, 11 April 1986). Also, Bernardo A. Huberman, “A Model for Dysfunctions in Smooth Pursuit Eye Movement,” preprint, Xerox Palo Alto Research Center, Palo Alto, California.
“THREE THINGS HAPPEN” Abraham. The basic introduction to the Gaia hypothesis—an imaginative dynamical view of how the earth’s complex systems regulate themselves, somewhat sabotaged by its deliberate anthropomorphism—is J. E. Lovelock, Gaia: A New Look at Life on Earth (Oxford: Oxford University Press, 1979).
RESEARCHERS INCREASINGLY RECOGNIZED A somewhat arbitrary selection of references on physiological topics (each with useful citations of its own): Ary L. Goldberger, Valmik Bhargava, and Bruce J. West, “Nonlinear Dynamics of the Heartbeat,” Physica 17D (1985), pp. 207–14. Michael C. Mackay and Leon Glass, “Oscillation and Chaos in Physiological Control Systems,” Science 197 (1977), p. 287. Mitchell Lewis and D. C. Rees, “Fractal Surfaces of Proteins,” Science 230 (1985), pp. 1163–65. Ary L. Goldberger, et al., “Nonlinear Dynamics in Heart Failure: Implications of Long-Wavelength Cardiopulmonary Oscillations,” American Heart Journal 107 (1984), pp. 612–15. Teresa Ree Chay and John Rinzel, “Bursting, Beating, and Chaos in an Excitable Membrane Model,” Biophysical Journal 47 (1985), pp. 357–66. A particularly useful and wide-ranging collection of other such papers is Chaos, Arun V. Holden, ed. (Manchester: Manchester University Press, 1986).
“A DYNAMICAL SYSTEM OF VITAL INTEREST” Ruelle, “Strange Attractors,” p. 48.
“IT’S TREATED BY PHYSICIANS” Glass.
“WE’RE AT A NEW FRONTIER” Goldberger.
MATHEMATICIANS AT THE COURANT INSTITUTE Peskin; David M. McQueen and Charles S. Peskin, “Computer-Assisted Design of Pivoting Disc Prosthetic Mitral Valves,” Journal of Thoracic and Cardiovascular Surgery 86 (1983), pp. 126–35.
A PATIENT WITH A SEEMINGLY HEALTHY HEART Cohen.
“THE BUSINESS OF DETERMINING” Winfree.
A STRONG SENSE OF GEOMETRY Winfree develops his view of geometric time in biological systems in a provocative and beautiful book, When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias (Princeton: Princeton University Press, 1987); a review article on the applications to heart rhythms is Arthur T. Winfree, “Sudden Cardiac Death: A Problem in Topology,” Scientific American 248 (May 1983), p. 144.
“I HAD A HEADFUL” Winfree.
“YOU GO TO A MOSQUITO” Winfree.
SHE REPORTED FEELING GREAT Strogatz; Charles A. Czeisler, et al., “Bright Light Resets the Human Circadian Pacemaker Independent of the Timing of the Sleep-Wake Cycle,” Science 233 (1986), pp.
667–70. Steven Strogatz, “A Comparative Analysis of Models of the Human Sleep-Wake Cycle,” preprint, Harvard University, Cambridge, Massachusetts.
HE HAD GAINED Winfree.
“WHEN MINES DECIDED” “Sudden Cardiac Death.”
TO DO SO, HOWEVER Ideker.
“THE CARDIAC EQUIVALENT” Winfree.
IDEKER’S IMMEDIATE INTENTION Ideker.
THEY USED TINY AGGREGATES Glass.
“EXOTIC DYNAMIC BEHAVIOR” Michael R. Guevara, Leon Glass, and Alvin Schrier, “Phase Locking, Period-Doubling Bifurcations, and Irregular Dynamics in Periodically Stimulated Cardiac Cells,” Science 214 (1981), p. 1350.
“MANY DIFFERENT RHYTHMS” Glass.
“IT IS A CLEAR INSTANCE” Cohen.
“PEOPLE HAVE MADE THESE WEIRD” Glass.
“DYNAMICAL THINGS ARE GENERALLY” Winfree.
“SYSTEMS THAT NORMALLY OSCILLATE” Leon Glass and Michael C. Mackay, “Pathological Conditions Resulting from Instabilities in Physiological Control Systems,” Annals of the New York Academy of Sciences316 (1979), p. 214.
“FRACTAL PROCESSES” Ary L. Goldberger, Valmik Bhargava, Bruce J. West, and Arnold J. Mandell, “Some Observations on the Question: Is Ventricular Fibrillation ‘Chaos,’” preprint.
“IS IT POSSIBLE” Mandell.
“WHEN YOU REACH AN EQUILIBRIUM” Mandell.
MANDELL OFFERED HIS COLLEAGUES Arnold J. Mandell, “From Molecular Biological Simplification to More Realistic Central Nervous System Dynamics: An Opinion,” in Psychiatry: Psychobiological Foundations of Clinical Psychiatry 3:2, J. O. Cavenar, et al., eds. (New York: Lippincott, 1985).
“THE UNDERLYING PARADIGM REMAINS” Ibid.
THE DYNAMICS OF SYSTEMS Huberman.
SUCH MODELS SEEMED TO HAVE Bernardo A. Huberman and Tad Hogg, “Phase Transitions in Artificial Intelligence Systems,” preprint, Xerox Palo Alto Research Center, Palo Alto, California, 1986. Also, Tad Hogg and Bernardo A. Huberman, “Understanding Biological Computation: Reliable Learning and Recognition,” Proceedings of the National Academy of Sciences 81 (1984), pp. 6871–75.
“ASTONISHING GIFT OF CONCENTRATING” Erwin Schrödinger, What Is Life? (Cambridge: Cambridge University Press, 1967), p. 82.
“IN PHYSICS WE HAVE DEALT” Ibid., p. 5.
CHAOS AND BEYOND
“WHEN I SAID THAT?” Ford.
“IN A COUPLE OF DAYS” Fox.
THE WORD ITSELF (Holmes) SIAM Review 28 (1986), p. 107; (Hao) Chaos (Singapore: World Scentific, 1984), p. i; (Stewart) “The Geometry of Chaos,” in The Unity of Science, Brookhaven Lecture Series, No. 209 (1984), p. 1; (Jensen) “Classical Chaos,” American Scientist (April 1987); (Crutchfield) private communication; (Ford) “Book Reviews,” International Journal of Theoretical Physics 25 (1986), No. 1.
TO HIM, THE OVERRIDING MESSAGE Hubbard.
TOO NARROW A NAME Winfree.
“IF YOU HAD A TURBULENT RIVER” Huberman.
“LET US AGAIN LOOK” Gaia, p. 125.
THOUGHTFUL PHYSICISTS P. W. Atkins, The Second Law (New York: W. H. Freeman, 1984), p. 179. This excellent recent book is one of the few accounts of the Second Law to explore the creative power of dissipation in chaotic systems. A highly individual, philosophical view of the relationships between thermodynamics and dynamical systems is Ilya Prigogine, Order Out of Chaos: Man’s New Dialogue With Nature (New York: Bantam, 1984).
GROWTH OF SUCH TIPS Langer. The recent literature on the dynamical snowflake is voluminous. Most useful are: James S. Langer, “Instabilities and Pattern Formation,” Reviews of Modern Physics (52) 1980, pp. 1–28; Johann Nittmann and H. Eugene Stanley, “Tip Splitting without Interfacial Tension and Dendritic Growth Patterns Arising from Molecular Anisotropy, Nature 321 (1986), pp. 663–68; David A. Kessler and Herbert Levine, “Pattern Selection in Fingered Growth Phenomena,” to appear in Advances in Physics.
IN THE BACK OF THEIR MINDS Gollub, Langer.
ODD-SHAPED TRAVELING WAVES An interesting example of this route to the study of pattern formation is P. C. Hohenberg and M. C. Cross, “An Introduction to Pattern Formation in Nonequilibrium Systems,” preprint, AT&T Bell Laboratories, Murray Hill, New Jersey.
IN ASTRONOMY, CHAOS EXPERTS Wisdom; Jack Wisdom, “Meteorites May Follow a Chaotic Route to Earth,” Nature 315 (1985), pp. 731–33, and “Chaotic Behavior and the Origin of the 3/1 Kirkwood Gap,” Icarus56 (1983), pp. 51–74.
STRUCTURES THAT REPLICATE THEMSELVES As Farmer and Packard put it: “Adaptive behavior is an emergent property which spontaneously arises through the interaction of simple components. Whether these components are neurons, amino acids, ants, or bit strings, adaptation can only occur if the collective behavior of the whole is qualitatively different from that of the sum of the individual parts. This is precisely the definition of nonlinear.” “Evolution, Games, and Learning: Models for Adaptation in Machines and Nature,” introduction to conference proceedings, Center for Nonlinear Studies, Los Alamos National Laboratory, May 1985.
“EVOLUTION IS CHAOS” “What Is Chaos?” p. 14.
“GOD PLAYS DICE” Ford.
“THE PROFESSION CAN NO LONGER” Structure, p. 5.
“BOTH EXHILARATING AND A BIT THREATENING” William M. Schaffer, “Chaos in Ecological Systems: The Coals That Newcastle Forgot,” Trends in Ecological Systems 1 (1986), p. 63.
“WHAT PASSES FOR FUNDAMENTAL” William M. Schaffer and Mark Kot, “Do Strange Attractors Govern Ecological Systems?” Bio-Science 35 (1985), p. 349.
SCHAFFER IS USING e.g., William M. Schaffer and Mark Kot, “Nearly One Dimensional Dynamics in an Epidemic,” Journal of Theoretical Biology 112 (1985), pp. 403–27.
“MORE TO THE POINT” Schaffer.
YEARS LATER, SCHAFFER LIVED Schaffer; also William M. Schaffer, “A Personal Hejeira,” unpublished.