On Chaos Theory

My dad, a fan of Chaos Theory, just sent me this interesting article about it and some of its history. Some of these things I already knew, but for the most part it was greatly revealing.
When I visit my parents home this winter, there will be a greater chance of me finally reading the pile of books he left on my desk a few months ago ;)

MATHEMATICS: CATASTROPHE THEORY, STRANGE ATTRACTORS, CHAOS

The following points are made by Nigel Calder (citation below):

1) Go out of Paris on the road towards Chartres and after 25 kilometers you will come to the Institut des Hautes Etudes Scientifiques at Bures-sur-Yvette. It occupies a quite small building surrounded by trees. Founded in 1958 in candid imitation of the Institute for Advanced Study in Princeton, it enables half a dozen lifetime professors to interact with 30 or more visitors in pondering new concepts in mathematics and theoretical physics. A former president, Marcel Boiteux, called it “a monastery where deep-sown seeds germinate and grow to maturity at their own pace.”

2) A recurring theme for the institute at Bures has been complicated behavior. In the 21st century this extends to describing how biological molecules — nucleic acids and proteins — fold themselves to perform precise functions. The mathematical monks in earlier days directed their attention towards physical and engineering systems that can often perform in complicated and unpredictable ways.

3) Catastrophe theory was invented at Bures-sur-Yvette in 1968. In the branch of mathematics concerned with flexible shapes, called topology, Rene Thom found origami-like ways of picturing abrupt changes in a system, such as the fracture of a girder or the capsizing of a ship. Changes that were technically catastrophic could be benign, for instance in the brain’s rapid switch from sleeping to waking. As the modes of sudden change became more numerous, the greater the number of factors affecting a system.

4) Fascinated colleagues included Christopher Zeeman at Warwick, who became Thom’s chief publicist. He and others also set out to apply catastrophe theory to an endless range of topics. From shock waves and the evolution of species, to economic inflation and political revolution, it seemed that no field of natural or social science would fail to benefit from its insights.

5) Thom himself blew the whistle to stop the folderol. “Catastrophe theory is dead,” he pronounced in 1997. “For as soon as it became clear that the theory did not permit quantitative prediction, all good minds… decided it was of no value.”

6) In an age of self-aggrandizement, Thom’s dismissal of his own theory set a refreshing example to others. But the catastrophe that overtook catastrophe theory has another lesson. Mathematics stands in relation to the rest of science like an exotic bazaar, full of pretty things but most of them useless to a visitor. Descriptions of logical relationships between imagined entities create wonderful worlds that never were or will be.

7) Mathematical scientists have to find the small selection of theorems that may describe the real world. Many decades can elapse in some cases before a particular item turns out to be useful. Then it becomes a jewel beyond price. Recent examples are the mathematical descriptions of subatomic particles, and of the motions of pieces of the Earth’s crust that cause earthquakes.

8) Sometimes the customer can carry a piece of mathematics home, only to find that it looks nice on the sideboard but doesn’t actually do anything useful. This was the failure of catastrophe theory. Thom’s origami undoubtedly provided mathematical metaphors for sudden changes, but it was not capable of predicting them.

9) When the subject is predictability itself, the relationship of science and mathematics becomes subtler. The next innovation at the leafy institute at Bures came in 1971. David Ruelle, a young Belgian-born permanent professor, and Floris Takens visiting from Groningen, were studying turbulence. If you watch a fast-moving river, you’ll see eddies and swirls that appear, disappear and come back, yet are never quite the same twice.

10) For understanding this not-quite-predictable behavior in an abstract, mathematical way, Ruelle and Takens wanted pictures. They were not sure what they would look like, but they had a curious name for them: “strange attractors”. Within a few years, many scientists’ computers would be doodling strange attractors on their monitors and initiating the genre of mathematical science called “chaos theory”.

11) To understand what attractors are, and in what sense they might be strange, you need first to look back to the pictures of Henri Poincare (1854-1912). He was France’s top theorist at the end of the 19th century. Wanting to visualize changes in a system through time, without getting mired in the details, he came up with a brilliantly simple method.

12) Put a dot in the middle of a blank piece of paper. It represents an unchanging situation. Not necessarily a static one, to be sure, because Poincare was talking about dynamical systems, but something in a steady state. It might be, for example, a population where births and deaths are perfectly balanced. All of the busy drama of courtship, childbirth, disease, accident, murder and senescence is then summed up in a geometric point. And around it, like the empty canvas that taunts any artist, the rest of the paper is an abstract picture of all possible variations in the behavior of the system. Poincare called it “phase space”. You can set it to work by putting a second dot on the paper. Because it is not in the middle, the new dot represents an unstable condition. So it cannot stay put, but must evolve into a curved line wandering across the paper. The points through which it passes are a succession of other unstable situations in which the system finds itself, with the passage of time. In the case of a population, the track that it follows depends on changes in the birth rate and death rate.

13) Considering the generality of dynamic systems, Poincare found that the curve often evolved into a loop that caught its own tail and continued on, around and around. It is not an actual loop, but a mathematical impression of a complicated system that has settled down into an endlessly repetitive cycle. A high birth rate may in theory increase a population until starvation sets in. That boosts the death rate and reverses the process. When there’s plenty to eat again, the birth rate recovers — and so on, ad infinitum.

14) Poincare also realized that systems coming from different starting conditions could finish up on the same loop in phase space, as if attracted to it by a latent preference in the type of dynamic behavior. A hypothetical population might commence with any combination of low or high rates of birth and death, and still finish up in the oscillation mentioned. The loop representing such a favored outcome is called an “attractor”.

15) In many cases the ultimate attractor is not a loop but the central dot representing a steady state. This may mean a state of repose, as when friction brings the swirling liquid in a stirred teacup to rest, or it may be the steady-state population where the birth rate and death rate always match. Whether they are loops or dots, Poincare attractors are tidy and you can make predictions from them.

16) By a strange attractor, Ruelle and Takens meant an untidy one that would capture the essence of the not-quite-predictable. Unknown to them an American meteorologist, Edward Lorenz, had already drawn a strange attractor in 1963, unaware of what its name should be. In his example it looked like a figure of eight drawn by a child taking a pencil around and around the same figure many times, but not at all accurately. The loop did not coincide from one circuit to the next, and you could not predict exactly where it would go next time.

Projection of trajectory of Lorenz system in phase space with "canonical" values of parameters r=28, ? = 10, b = 8/3

Projection of trajectory of Lorenz system in phase space with “canonical” values of parameters r=28, ? = 10, b = 8/3

17) When mathematicians woke up to this convergence of research in France and the USA, they proclaimed the advent of “chaos”. The strange attractor was its emblem. An irony is that Poincare himself had discovered chaos in the late 1880s, when he was shocked to find that the motions of the planets are not exactly predictable. But as he didn’t use an attention-grabbing name like chaos, or draw any pictures of strange attractors, the subject remained in obscurity for more than 80 years, nursed mainly by mathematicians in Russia.

18) Chaos in its contemporary mathematical sense acquired its name from James Yorke of Princeton, in a paper published in 1975. Assisting in the relaunch of the subject was Robert May, also at Princeton, who showed that a childishly simple mathematical equation could generate extremely complicated patterns of behavior. And in the same year, Mitchell Feigenbaum at the Los Alamos National Laboratory in New Mexico discovered a magic number. This is delta, 4.669201…, and it keeps cropping up in chaos, as pi does in elementary geometry. Rhythmic variations can occur in chaotic systems, and then switch to a rhythm at twice the rate. The Feigenbaum number helps to define the change in circumstances — the speed of a stream for example — needed to provoke transitions from one rhythm to the next.

19) Here was evidence of latent orderliness that distinguishes certain kinds of erratic behavior from mere chance. “Chaos is not random: it is apparently random behavior resulting from precise rules,” explained lan Stewart of Warwick. “Chaos is a cryptic form of order.” During the next 20 years, the mathematical idea of chaos swept through science like a tidal wave. It was the smart new way of looking at everything from fluid dynamics to literary criticism. Yet by the end of the century the subject was losing some of its glamor.

20) Exhibit A, for anyone wanting to proclaim the importance of chaos, was the weather. Indeed it set the trend, with Lorenz’s unwitting discovery of the first strange attractor. That was a by-product of his experiments on weather forecasting by computer at the beginning of the 1960s. As an atmospheric scientist of mathematical bent at the Massachusetts Institute of Technology, Lorenz used a very simple simulation of the atmosphere by numbers, and computed changes at a network of points.

21) He was startled to find that successive runs from the same starting point gave quite different weather predictions. Lorenz traced the reason. The starting points were not exactly the same. To launch a new calculation he was using rounded numbers from a previous calculation. For example, 654321 became 654000. He had assumed, wrongly, that such slight differences were inconsequential. After all, they corresponded to mere millimeters per second in the speed of the wind.

22) This was the “Butterfly Effect”. Lorenz’s computer told him that the flap of a butterfly’s wings in Brazil might stir up a tornado in Texas. A mild interpretation said that you would not be able to forecast next week’s weather very accurately because you couldn’t measure today’s weather with sufficient precision. But even if you could do so, and could lock up all the lepidoptera, the sterner version of the Butterfly Effect said that there was enough unpredictable turbulence in the smallest cloud to produce chance variations of a greater degree.

23) The dramatic inference was that the weather would do what it damn well pleased. It was inherently chaotic and unpredictable. The Butterfly Effect was a great comfort to meteorologists trying to use the primitive computers of the 1960s for long-range weather forecasts. “We certainly hadn’t been successful at doing that anyway,” Lorenz said, “and now we had an excuse.”

Adapted from: Nigel Calder: Magic Universe: The Oxford Guide to Modern Science. Oxford University Press 2003, p.133. More information at: http://www.amazon.com/exec/obidos/ASIN/0198507925/scienceweek

Taken from ScienceWeek

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