• Choose a category
• All
• Classic-Fiction

By Root 822 0

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.

REVOLUTION

THE