Science at the Frontier (1992) / Chapter Skim
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3 Dynamical Systems: When the Simple Is Complex: New Mathematical Approaches to Learning About the Universe
3 Dynamical Systems: When the Simple Is Complex: New Mathematical Approaches to Learning About the Universe
Pages 45-65
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From page 45... ...
, is not so difficult, and linear models are adequate for many practical tasks, such as building a bridge to support a given load without the bridge vibrating so hard that it shakes itself apart. But most dynamical systems including all the really interesting problems, from guidance of satellites and the turbulence of boiling water to the dynamics of Earth's atmosphere and the electrical activity of the brain involve nonlinear processes and are far harder to understand.
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From page 46... ...
In the 1970s, mathematician James Yorke from the University of Maryland's Institute for Physical Science and Technology used the word chaos to describe the apparently random behavior of a certain class of nonlinear systems (York and Li, 1975~. The word can be misleading: it refers not to a complete lack of order but to apparently random behavior with nonetheless decipherable pattern.
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From page 47... ...
Louis; and Curt McMullen from Princeton, currently at the University of California, Berkeley.) Mathematicians laid the foundations for chaos and dynamical systems; their continuing involvement in such a highly visible field has raised questions about the use of computers as an experimental tool for mathematics and about whether there is a conflict between mathematicians' traditional dedication to rigor and proof, and enticing color pictures on the computer screen or in a coffee-table book.
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From page 48... ...
discovered and explored what is now known as Julia sets, seen today as characteristic examples of chaotic behavior. In the United States George Birkhoff developed many fundamental techniques; in 1913 he gave the first correct proof of one of Poincare's major conjectures.
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From page 49... ...
Sensitivity to Initial Conditions Sensitivity to initial conditions is what chaos is all about: a chaotic system is one that is sensitive to initial conditions. In his talk McMullen described a mathematical example, "the simplest dynamical system in the quadratic family that one can study," iteration of the polynomial x2 + c when c = 0.
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From page 50... ...
"One way to think of the Julia set is as the boundary between those values of z that iterate to infinity, and those that do not," said McMullen. The Julia set for the simple quadratic polynomial Z2 + C when c = 0, is the unit circle, the circle with radius 1: values of z outside the circle iterate to infinity; values inside go to 0.
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From page 52... ...
Traditionally the horizontal axis represents the real numbers, and so the number x tells how far right or left z is. The vertical axis represents imaginary numbers, so the number y tells how far up or down z is.
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From page 53... ...
The single line at the top is formed of points that are attracting fixed points for the polynomials x2 + c: one such fixed point for each value of c. Where the line splits, the values produced by iteration bounce between two points (called an attractive cycle of period two)
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From page 54... ...
The single line at the top is formed of points that are attracting fixed points. Where the line splits, the values produced by iteration bounce between two points; where they split again they bounce between four points, and so on.
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From page 55... ...
In their model, the doubling time is approximately 3 days. Thus, roughly speaking, if you have completely lost control of your system after 25 doubling times, then it is very realistic to think that the fluttering of the wings of a butterfly can have an absolutely determining effect on the weather 75 days later.
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From page 56... ...
"Any time you have a system that doubles uncertainties, the shadowing lemma of mathematics applies," Hubbard said. "It says that you cannot tell whether a given system is deterministic or randomly perturbed at a particular scale, if you can only observe it at that scale." The simplest example of such a system, he said, is angle doubling.
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From page 57... ...
The largest view of the Mandelbrot set looks like a symmetric beetle or gingerbread man, with nodes or balls attached (Figure 3.3~; looked at more closely, these balls or nodes turn out to be miniature copies of the "parent" Mandelbrot set, each in a different and complex setting (Figure 3.4~. In between these nodes, a stunning array of geometrical shapes emerges: beautiful, sweeping, intricate curves and arcs and swirls and scrolls and curlicues, shapes that evoke marching elephants, prancing seahorses, slithering insects, and coiled snakes, lightning flashes and showers of stars and fireworks, all emanating from the depths of the screen.
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From page 58... ...
The Mandelbrot set some mathematicians prefer the term quadratic bifurcation locus summarizes all the Julia sets for quadratic polynomials. Julia sets can be roughly divided into those that are connected and those that are not.
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From page 59... ...
The "computer program" that creates a human being the DNA in the genome contains about 4 billion bits (units of information)
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From page 60... ...
Hubbard sees the relatively tiny amount of information in the human gene as "a marvelous message of hope.... A large part of why I did not become a biologist was precisely because I felt that although clearly it was the most interesting subject in the world, there was almost no hope of ever understanding anything in any adequate sense.
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From page 61... ...
McMullen expressed a similar view: "The computer is a fantastic vehicle for transforming mathematical reality, for revealing something that was hitherto just suspended within the consciousness of a single person. I find it indispensable.
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From page 62... ...
"People who make statements suggesting that chaos is the third great revolution in 20th-century science, following quantum mechanics and relativity, are quite easy targets for a comeback
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From page 63... ...
I mean once I really think I understand it, then I can sit down and try to prove it." Part of the difficulty, Devaney suggested, is the very availability of the computer as an experimental tool, with which mathematicians can play, varying parameters and seeing what happens. "With the computer now finally mathematics has its own essentially experimental component," Devaney told fellow scientists at the symposium.
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From page 64... ...
Beyond that there is a tremendous amount of accessible mathematics." Yet these concerns seem relatively minor compared to the insights that have been achieved. It may well distress mathematicians that claims for chaos have been exaggerated, that the field has inspired some less than rigorous work, and that the public may marvel at the superficial beauty of the pictures while ignoring the more profound, intrinsic beauty of their mathematical meaning, or the beauty of other, less easily illustrated, mathematics.
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From page 65... ...
Part I: Ordinary Differential Equations. Springer-Verlag, New York.
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