## 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 A few provided unpublished written recollections.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:

Günter Ahlers

F.Tito Arecchi

Michael Barnsley

Lennart Bengtsson

William D. Bonner

Robert Buchal

William Burke

David Campbell

Peter A. Carruthers

Richard J. Cohen

James Crutchfield

Predrag Cvitanović

Minh Duong-van

Freeman Dyson

Jean-Pierre Eckmann

Fereydoon Family

J. Doyne Farmer

Mitchell J. Feigenbaum

Joseph Ford

Ronald Fox

Robert Gilmore

Leon Glass

James Glimm

Ary L. Goldberger

Jerry P. Gollub

Ralph E. Gomory

Stephen Jay Gould

John Guckenheimer

Brosl Hasslacher

Michel Hénon

Douglas R. Hofstadter

Pierre Hohenberg

Frank Hoppensteadt

Hendrik Houthakker

John H. Hubbard

Bernardo Huberman

Raymond E. Ideker

Erica Jen

Roderick V. Jensen

Leo Kadanoff

Donald Kerr

Joseph Klafter

Thomas S. Kuhn

Mark Laff

Oscar Lanford

James Langer

Joel Lebowitz

Cecil E. Leith

Herbert Levine

Albert Libchaber

Edward N. Lorenz

Willem Malkus

Benoit Mandelbrot

Arnold Mandell

Syukuro Manabe

Arnold J. Mandell

Philip Marcus

Paul C. Martin

Robert M. May

Francis C. Moon

Jürgen Moser

David Mumford

Michael Nauenberg

Norman Packard

Heinz-Otto Peitgen

Charles S. Peskin

James Ramsey

Peter H. Richter

Otto Rössler

David Ruelle

William M. Schaffer

Stephen H. Schneider

Christopher Scholz

Robert Shaw

Michael F. Shlesinger

Yasha G. Sinai

Steven Smale

Edward A. Spiegel

H. Bruce Stewart

Steven Strogatz

Harry Swinney

Tomas Toffoli

Felix Villars

William M. Visscher

Richard Voss

Bruce J. West

Robert White

Gareth P. Williams

Kenneth G. Wilson

Arthur T. Winfree

Jack Wisdom

Helena Wisniewski

Steven Wolfram

J. Austin Woods

James A. Yorke

**PROLOGUE**

__L____OS ALAMOS__ Feigenbaum, Carruthers, Campbell, Farmer, Visscher, Kerr, Hasslacher, Jen.

“__I____ UNDERSTAND YOU’RE__” Feigenbaum, Carruthers.

__G____OVERNMENT 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.

__T____HAT REALIZATION__ Mandelbrot, Ramsey; Wisdom, Marcus; Alvin M. Saperstein, “Chaos—A Model for the Outbreak of War,” *Nature* 309 (1984), pp. 303–5.

“__F____IFTEEN YEARS AGO__” Shlesinger.

__JUST THREE THINGS__ Shlesinger.

__T____HIRD GREAT REVOLUTION__ Ford.

“__R____ELATIVITY ELIMINATED__” Joseph Ford, “What Is Chaos, That We Should Be Mindful of It?” preprint, Georgia Institute of Technology, p. 12.

__T____HE 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**

__T____HE 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.

__T____HEY 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.

“__I____T WOULD EMBRACE__” Pierre Simon de Laplace, *A Philosophical Essay on Probabilities* (New York: Dover, 1951).

“__T____HE BASIC IDEA__” Winfree.

“__T____HAT’S THE KIND OF RULE__” Lorenz.

__S____UDDENLY 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.

__T____HAT FIRST DAY__ Lorenz; also, “On the Prevalence,” p. 56.

“__W____E 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.

__V____ON NEUMANN IMAGINED__ Dyson.

__VAST AND EXPENSIVE BUREAUCRACY__ Bonner, Bengtsson, Woods, Leith.

__F____ORECASTS OF ECONOMIC__ Peter B. Medawar, “Expectation and Prediction,” in Pluto’s Republic (Oxford: Oxford University Press, 1982), pp. 301–4.

__T____HE 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.

“__P____REDICTION, NOTHING__” Lorenz, White.

__THERE MUST BE A LINK__ “The Mechanics of Vacillation.”

__F____OR 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.”

“__T____HE 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.

“__W____E MIGHT HAVE TROUBLE__” Ibid., p. 410.

__L____ORENZ 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.

__T____HREE 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).

“__E____D, 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.

**REVOLUTION**

__T____HE 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.

__E____XPERIMENTALISTS CARRY OUT__ Tension, p. 229.

__I____N BENJAMIN FRANKLIN’S__ STRUCTURE, pp. 13–15.

“__U____NDER NORMAL CONDITIONS__ TENSION, p. 234.

__A____ PARTICLE PHYSICIST__ Cvitanović

__T____OLSTOY__ 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.

“__I____T’S MASOCHISM__ Richter.

__T____HESE 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.

__T____HE HEART OF CHAOS__ Alan Wolf, “Simplicity and Universality in the Transition to Chaos,” *Nature* 305 (1983), p. 182.

__C____HAOS NOW PRESAGES__ Joseph Ford, “What is Chaos, That We Should Be Mindful of It?” preprint, Georgia Institute of Technology, Atlanta.

__R____EVOLUTIONS DO NOT__ “What Are Scientific Revolutions?” p. 23.

“__I____T IS RATHER AS IF__” Structure, p. 111.

__T____HE LABORATORY MOUSE__ Yorke and others.

__W____HEN ARISTOTLE LOOKED__ “What Are Scientific Revolutions?” pp. 2–10.

“__I____F 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.

__T____HAT 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.

__Y____ET, 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.

__S____PACE 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.”

__F____RENCH ASTRONOMER__ Hénon.

__J____APANESE ELECTRICAL ENGINEER__ Ueda. 45 A YOUNG PHYSICIST Fox.

__S____MALE__ 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.

__T____HE 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 Intelligencer*6:2, pp. 21–27.

__W____HEN HE RETURNED__ Smale.

__A L____ETTER 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.

__R____OBUST AND STRANGE__ Smale; “On How I Got Started.”

__I____T WAS JUST A VACUUM TUBE__ van der Pol described his work in Nature 120 (1927), pp. 363–64.

“__O____FTEN AN IRREGULAR NOISE__” Ibid.

__T____O MAKE A SIMPLE__ Smale’s definitive mathematical exposition of this work is “Differentiable Dynamical Systems,” *Bulletin of the* A*merican Mathematical Society* 1967, pp. 747–817 (also in *The Mathematics of Time*, pp. 1–82).

__T____HE PROCESS MIMICS__ Rössler.

__B____UT FOLDING__ Yorke.

__I____T WAS A GOLDEN AGE__ Guckenheimer, Abraham.

“__I____T’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.

__V____OYAGER HAD MADE__ Ingersoll; also, Andrew P. Ingersoll, “Order from Chaos: The Atmospheres of Jupiter and Saturn,” Planetary Report 4:3, pp. 8–11.

“__Y____OU SEE THIS__” Marcus.

“__G____EE, WHAT ABOUT__” Marcus.

**LIFE’S UPS AND DOWNS**

__R____AVENOUS 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).

__T____HE 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.

__B____Y THE__ 1950s May.

__R____EFERENCE 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.

__H____E 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.

__L____ORENZ’S PAPER__ Yorke.

“__F____ACULTY 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.

__H____E GAVE A COPY__ Yorke, Smale.

“__I____F 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.

“__I____T 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.

“__T____HE FIRST MESSAGE__” Yorke.

__Y____ORKE’S PAPER__ Written with his student Tien-Yien Li. “Period Three Implies Chaos,” *American Mathematical Monthly* 82 (1975), pp. 985–92.

__M____AY CAME TO BIOLOGY__ May.

“__W____HAT THE CHRIST__” May; it was this seemingly unanswerable question that drove him from analytic methods to numerical experimentation, meant to provide intuition, at least.

__S____TARTLING THOUGH IT WAS__ Yorke.

__A. N. S____ARKOVSKII__ “Coexistence of Cycles of a Continuous Map of a Line into Itself,” *Ukrainian Mathematics Journal* 16 (1964), p. 61.

__S____OVIET MATHEMATICIANS AND PHYSICISTS__ Sinai, personal communication, 8 December 1986.

__SOME____ W____ESTERN CHAOS EXPERTS__ e.g., Feigenbaum, Cvitanović.

__T____O SEE DEEPER__ Hoppensteadt, May.

__T____HE FEELING OF ASTONISHMENT__ Hoppensteadt.

__W____ITHIN ECOLOGY__ May.

__N____EW 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.

__T____HE WORLD WOULD BE__ “Simple Mathematical Models,” p. 467.

“__T____HE 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.

__W____ASSILY__ LEONTIEF Quoted in Fractal Geometry, p. 423.

__I____NTRODUCED FOR A LECTURE__ Woods Hole Oceanographic Institute, August 1985.

__BORN IN____ W____ARSAW__ Mandelbrot.

__B____OURBAKI__ 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.

__T____HE FIELD DEVELOPS__ Peitgen.

__PIONEER-BY–NECESSITY__ “Second Stage,” p. 5.

__T____HIS 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.

__T____HE JOSEPH EFFECT__ Fractal Geometry, p. 248.

__C____LOUDS ARE NOT SPHERES__ Ibid., p. 1, for example.

__W____ONDERING ABOUT COASTLINES__ Ibid., p. 27.

__T____HE PROCESS OF ABSTRACTION__ Ibid., p. 17.

“__T____HE NOTION__” Ibid., p. 18.

__O____NE WINTRY AFTERNOON__ Mandelbrot.

__T____HE EIFFEL TOWER__ Fractal Geometry, p. **131,** and “On Fractal Geometry,” p. 1663. 102 ORIGINATED BY MATHEMATICIANS F. Hausdorff and A. S. Besicovich.

“__T____HERE WAS A LONG HIATUS__” Mandelbrot.

__I____N 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.

“__I____T’S A SINGLE MODEL__” Scholz.

“__I____N 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).

__T____HE NETWORK OF SPECIAL FIBERS__ Goldberger, West.

__S____EVERAL 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 Journal*48 (1985), p. 525.

__W____HEN 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.

__H____IS NAME APPEARED__ I. Bernard Cohen, *Revolution in Science* (Cambridge, Mass.: Belknap, 1985), p. 46.

“__O____F COURSE, HE IS A BIT__” Mumford.

“__H____E 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.

“__M____ANDELBROT DIDN’T HAVE EVERYBODY’S__” Richter.

“__T____HE POLITICS AFFECTED__” Mandelbrot.

__E____XXON’S HUGE RESEARCH FACILITY__ Klafter.

__O____NE MATHEMATICIAN TOLD FRIENDS__ Related by Huberman.

“__W____HY IS IT THAT__” “Freedom, Science, and Aesthetics,” in *Schönheit im Chaos*, p. 35.

“__T____HE PERIOD HAD NO SYMPATHY__” John Fowles, *A Maggot* (Boston: Little, Brown, 1985), p. 11.

“__W____E 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.

__B____UT 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?”

**STRANGE ATTRACTORS**

__T____HE 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.

__T____HERE 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.”

__T____HIS 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.

“__T____HAT’S TRUE__” Swinney.

__I____N 1973 SWINNEY__ Swinney, Gollub.

“__I____T WAS A STRING-AND–SEALING-WAX__” Dyson.

“__S____O WE READ THAT__” Swinney.

__W____HEN THEY BEGAN REPORTING__ Swinney, Gollub.

“__T____HERE WAS THE TRANSITION__” Swinney.

__E____XPERIMENT 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.

__D____AVID RUELLE SOMETIMES SAID__ Ruelle. 132.

“__A____LWAYS NONSPECIALISTS FIND__” Ruelle.

__H____E 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.

“__D____ID YOU EVER ASK____ G____OD__” “Strange Attractors,” p. 131.

“__T____AKENS HAPPENED__” Ruelle.

“__S____OME MATHEMATICIANS IN__ CALIFORNIA” Ralph H. Abraham and Christopher D. Shaw, Dynamics: *The Geometry of Behavior* (Santa Cruz: Aerial: 1984).

“__I____T ALWAYS BOTHERS ME__” Richard P. Feynman, *The Character of Physical Law* (Cambridge, Mass.: The M.I.T. Press, 1967), p. 57.

__D____AVID RUELLE SUSPECTED__ Ruelle.

__T____HE REACTION OF THE SCIENTIFIC PUBLIC__ “Turbulent Dynamical Systems,” p. 275.

__E____DWARD__ LORENZ HAD ATTACHED “Deterministic Nonperiodic Flow,” p. 137.

“__I____T IS DIFFICULT TO RECONCILE__ Ibid., p. 140.

__H____E WENT TO VISIT LORENZ__ Ruelle.

“__D____ON’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.

__T____HE 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).

__I____S THE SOLAR SYSTEM__ Wisdom.

“__T____O 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.

__A____T THE OBSERVATORY__ Hénon.

“__I____, TOO, WAS CONVINCED__” Hénon.

“__H____ERE COMES THE SURPRISE__” “The Applicability,” p. 76.

“__BUT THE MATHEMATICAL APPROACH__” Ibid., p. 79.

__A VISITING PHYSICIST__ Yves Pomeau.

“__SOMETIMES ASTRONOMERS ARE FEARFUL__” Hénon.

__O____THERS ASSEMBLED MILLIONS__ Ramsey.

“__I____ HAVE NOT SPOKEN__” “Strange Attractors,” p. 137.

**UNIVERSALITY**

“__Y____OU 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.”

__W____HEN FEIGENBAUM CAME TO LOS ALAMOS__ Feigenbaum, Carruthers, Cvitanović, Campbell, Farmer, Visscher, Kerr, Hasslacher, Jen.

“__I____F YOU HAD SET UP__” Carruthers.

__T____HE MYSTERY OF THE UNIVERSE__ Feigenbaum.

__O____CCASIONALLY AN ADVISOR__ Carruthers.

__A____S KADANOFF VIEWED__ Kadanoff.

“__T____HE CEASELESS MOTION__” Gustav Mahler, letter to Max Marschalk.

“__W____ITH 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.

__T____HIS 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.

__T____O____ M____ETROPOLIS____, S____TEIN, AND____ S____TEIN__ 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.

__D____OES A CLIMATE EXIST__ “The Problem of Deducing the Climate from the Governing Equations,” Tellus 16 (1964), pp. 1–11.

__THE____ W____HITE____ E____ARTH CLIMATE__ Manabe.

__H____E KNEW NOTHING OF LORENZ__ Feigenbaum.

__O____DDLY__ May.

__T____HE 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.

“__T____HE WHOLE TRADITION OF PHYSICS__” Feigenbaum.

__H____IS FRIENDS SPECULATED__ Cvitanović.

__S____UDDENLY YOU COULD SEE__ Ford.

__P____RIZES AND AWARDS__ The MacArthur fellowship; the 1986 Wolf Prize in physics.

“__F____EIGENBAUMOLOGY__” Dyson.

“__I____T WAS A VERY HAPPY__” Gilmore.

__B____UT ALL THE WHILE__ Cvitanović.

__WORK BY____ O____SCAR____ E. L____ANFORD__ 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.

“__S____IR, DO YOU MEAN__” Feigenbaum; ”*The Discovery of Universality*,” p. 17.

__I____N THE SUMMER OF__ 1977 Ford, Feigenbaum, Lebowitz.

“__M____ITCH HAD SEEN UNIVERSALITY__” Ford.

“__S____OMETHING DRAMATIC HAPPENED__” Feigenbaum.

**THE EXPERIMENTER**

“__A____LBERT IS GETTING MATURE__” Libchaber, Kadanoff.

__H____E SURVIVED THE WAR__ Libchaber.

“__H____EUUM 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.

__T____HE LABORATORY OCCUPIED__ Libchaber, Feigenbaum.

“__S____CIENCE WAS CONSTRUCTED__” Libchaber.

“__B____UT 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.

“__T____HIS PICTURE OF STRANDS__” Ibid., p. 16.

“__T____HE INEQUALITIES__” Ibid., p. 39.

“__I____T 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.

“__F____EW 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.

__I____T WAS SO SENSITIVE__ Campbell.

“__I____T WAS CLASSICAL PHYSICS__” Libchaber.

__N____OW, HOWEVER, A NEW FREQUENCY__ Libchaber and Maurer, 1980 and 1981. Also Cvitanović’s introduction gives a lucid summary.

“__T____HE NOTION THAT THE ACTUAL__” Hohenberg.

“__T____HEY STOOD AMID THE SCATTERED__” Feigenbaum, Libchaber.

“__Y____OU 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.

__T____O 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.

__I____N 1980 A EUROPEAN GROUP__ P. Collet, J.–P. Eckmann, and H. Koch, “Period Doubling Bifurcations for Families of Maps on R^{n},” *Journal of Statistical Physics* 25 (1981), p. 1.

“__A____ PHYSICIST WOULD ASK ME__” Libchaber.

**IMAGES OF CHAOS**

__M____ICHAEL BARNSLEY MET__ Barnsley.

__R____UELLE SHUNTED IT BACK__ Barnsley.

__J____OHN____ H____UBBARD, AN____ A____MERICAN__ 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.

“__N____OW, FOR EQUATIONS__” “Julia Sets and the Mandelbrot Set,” p. 170.

__H____E 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.

__T____HE____ M____ANDELBROT 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.

“__Y____OU OBTAIN AN INCREDIBLE VARIETY__ “Julia Sets and the Mandelbrot Set,” p. 161.

__I____N 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.

__A____S HE TRIED CALCULATING__ Mandelbrot; *The Beauty of Fractals*.

__M____ANDELBROT STARTED WORRYING__ Mandelbrot.

__NO TWO PIECES ARE “TOGETHER”__ Hubbard.

“__E____VERYTHING WAS VERY GEOMETRIC__” Peitgen.

__A____T CORNELL, MEANWHILE__ Hubbard.

__R____ICHTER HAD COME TO COMPLEX SYSTEMS__ Richter.

“__I____N A BRAND NEW AREA__” Peitgen.

“__R____IGOR IS THE STRENGTH__” Peitgen.

__F____RACTAL 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.

“__N____OBODY CAN SAY__” Yorke, remarks at Conference on Perspectives in Biological Dynamics and Theoretical Medicine, National Institutes of Health, Bethesda, Maryland, 10 April 1986.

__T____YPICALLY, 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.

“__P____ERHAPS WE SHOULD BELIEVE__” *The Beauty of Fractals*, p. 136.

__W____HEN 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.

“__I____F THE IMAGE IS COMPLICATED__” Barnsley.

“__T____HERE IS NO RANDOMNESS__” Hubbard.

“__R____ANDOMNESS IS A RED__” Barnsley.

**THE DYNAMICAL SYSTEMS COLLECTIVE**

__S____ANTA____ C____RUZ__ 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).

__H____E DID NOT KNOW__ Shaw.

__W____ILLIAM____ B____URKE____, a S____ANTA____ C____RUZ 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.

__T____HE ORIGINAL PLANS__ Farmer, Crutchfield.

__B____Y BUILDING UP__ Shaw, Crutchfield, Burke.

__A____ FEW MINUTES LATER__ Shaw.

“__A____LL YOU HAVE TO DO__” Abraham.

__D____OYNE____ F____ARMER__ 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.

__P____HYSICS AT__ SANTA CRUZ Burke, Farmer, Crutchfield.

“__GIZMO-ORIENTED__” Shaw.

__F____ORD HAD ALREADY DECIDED__ Ford.

__T____HEY 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.

“__W____HEN ONE MEETS THE CONCEPT__” Ibid., p. 13.

__N____ORMAN____ P____ACKARD WAS READING__ Packard.

__I____N____ D____ECEMBER__ 1977 Shaw.

__W____HEN____ L____ORENZ WALKED INTO THE ROOM__ Shaw, Farmer.

__H____E FINALLY MAILED HIS PAPER__ “Strange Attractors, Chaotic Behavior, and Information Flow.”

__A. N. K____OLMOGOROV AND YASHA SINAI__ Sinai, private communication.

__A____T THE PINNACLE__ Packard.

“__Y____OU DON’T SEE SOMETHING__” Shaw.

“__I____T’S A SIMPLE EXAMPLE__” Shaw.

__SYSTEMS THAT THE____ S____ANTA____ C____RUZ 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.

__O____NE IMPORTANT VARIABLE__ The Dripping Faucet, p. 4.

“__A STATE-OF–THE-ART COMPUTER CALCULATION__” Ibid.

__A “PSEUDOCOLLOQUIUM”__ Crutchfield.

“__IT TURNS OUT__” Shaw.

“__W____HEN 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.

“__G____OD, WE’RE STILL__” Crutchfield.

__SOME PROFESSORS DENIED__ e.g., Nauenberg.

“__W____E 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.

__L____ANFORD LISTENED POLITELY__ Farmer.

“__I____T WAS MY NAIVETÉ__” Farmer.

“__A____UDIOVISUAL AIDS__” Shaw.

__O____NE DAY____ B____ERNARDO____ H____UBERMAN__ crutchfield, huberman.

“__I____T 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.

__F____ARMER WAS ANGERED__ Crutchfield.

__C____LIMATE SPECIALISTS__ This is a continuing debate in the journal Nature, for example.

__E____CONOMISTS ANALYZING STOCK MARKET__ Ramsey.

__F____RACTAL 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.

**INNER RHYTHMS**

__H____UBERMAN 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.

“__T____HREE 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).

__R____ESEARCHERS 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.

“__I____T’S TREATED BY PHYSICIANS__” Glass.

“__W____E’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,” J*ournal of Thoracic and Cardiovascular Surgery* 86 (1983), pp. 126–35.

__A____ PATIENT WITH A SEEMINGLY HEALTHY HEART__ Cohen.

“__T____HE 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.

“__Y____OU 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.

__H____E HAD GAINED__ Winfree.

“__W____HEN____ M____INES DECIDED__” “Sudden Cardiac Death.”

__T____O DO SO, HOWEVER__ Ideker.

“__THE CARDIAC EQUIVALENT__” Winfree.

__I____DEKER’S IMMEDIATE INTENTION__ Ideker.

__T____HEY USED TINY AGGREGATES__ Glass.

“__E____XOTIC 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.

“__M____ANY DIFFERENT RHYTHMS__” Glass.

“__I____T IS A CLEAR INSTANCE__” Cohen.

“__P____EOPLE HAVE MADE THESE WEIRD__” Glass.

“__D____YNAMICAL THINGS ARE GENERALLY__” Winfree.

“__S____YSTEMS 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 Sciences*316 (1979), p. 214.

“__F____RACTAL PROCESSES__” Ary L. Goldberger, Valmik Bhargava, Bruce J. West, and Arnold J. Mandell, “Some Observations on the Question: Is Ventricular Fibrillation ‘Chaos,’” preprint.

“__I____S IT POSSIBLE__” Mandell.

“__W____HEN YOU REACH AN EQUILIBRIUM__” Mandell.

__M____ANDELL 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).

“__T____HE UNDERLYING PARADIGM REMAINS__” Ibid.

__T____HE DYNAMICS OF SYSTEMS__ Huberman.

__S____UCH 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 I*s *Life?* (Cambridge: Cambridge University Press, 1967), p. 82.

“__I____N PHYSICS WE HAVE DEALT__” Ibid., p. 5.

**CHAOS AND BEYOND**

“__W____HEN I SAID THAT__?” Ford.

“__I____N A COUPLE OF DAYS__” Fox.

__T____HE 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.

__T____O HIM, THE OVERRIDING MESSAGE__ Hubbard.

__T____OO NARROW A NAME__ Winfree.

“__I____F YOU HAD A TURBULENT RIVER__” Huberman.

“__L____ET US AGAIN LOOK__” Gaia, p. 125.

__T____HOUGHTFUL 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).

__G____ROWTH 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*.

__I____N THE BACK OF THEIR MINDS __Gollub, Langer.

__O____DD-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.

__I____N 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,” *Icarus*56 (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.

“__E____VOLUTION IS CHAOS__” “What Is Chaos?” p. 14.

“__G____OD 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.

“__W____HAT PASSES FOR FUNDAMENTAL__” William M. Schaffer and Mark Kot, “Do Strange Attractors Govern Ecological Systems?” *Bio-Science* 35 (1985), p. 349.

__S____CHAFFER 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.

“__M____ORE TO THE POINT__” Schaffer.

__Y____EARS LATER, SCHAFFER LIVED__ Schaffer; also William M. Schaffer, “A Personal Hejeira,” unpublished.