Chaos - James Gleick [160]
“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