MATHEMATICS: CATASTROPHE THEORY, STRANGE ATTRACTORS, CHAOS
ScienceWeek
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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.
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.
1
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
ScienceWeek
http://www.scienceweek.com
--
Best,
Frederick Martin McNeill
Poway, California, United States of America
mmcneill@fuzzysys.com
http://www.fuzzysys.com
http://members.cox.net/fmmcneill/
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Phrase of the week :
"There is no doubt that great revolutions of human scientific
thought will occur in the next century, and in the century after
that, and in thousands of centuries afterward. So which of our
current pet scientific dogmas will be among the first washed away
by new facts and sudden clarities?" -- Anonymous
)))Snort!)
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