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It turns out that an eerie type of chaos can lurk just behind a facade of order - and yet,
deep inside the chaos lurks an even eerier type of order
- Douglas Hofstadter.
This isn't right. This isn't even wrong
-Wolfgang Pauli
I will put Chaos into fourteen lines
And keep him there; and let him thence escape
If he be lucky; let him twist and ape
Flood, fire, and demon - his adroit designs
Will strain to nothing in the strict confines
Of this sweet Order, where, in pious rape,
I hold his essence and amorphous shape,
Till he with Order mingles and combines.
Past are the hours, the years, of our duress,
His arrogance, our awful servitude: I have him.
He is nothing more nor less
Than something simple not yet understood;
I shall not even force him to confess;
Or answer. I will only make him good.
Edna St. Vincent Millay
The Butterfly Effect
The popular name given to the exquisite sensitivity to initial conditions of chaotic
nonlinear systems. The term was used by Dr. Edward Lorenz of MIT in a 1972 lecture
entitle, Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado
in Texas? The point of this seemingly facetious question is that in chaotic system a
minuscule perturbation can have large and
unanticipated effects. This possibility wreaks havoc with traditional linear ideas of
causality based on single-cause/single effect mechanisms and proportional responses.
Edward N. Lorenz is recognized world-wide not only as one of the outstanding theoretical
meteorologists of our day, but also as a pioneer in the new major area of scientific study
known as "deterministic chaos," which has applications in many fields ranging
from pure mathematics to physics, engineering, chemistry, biology, economics and geology
as well as his own field of atmospheric science.
As a theoretical meteorologist, he refined the concept of available potential energy and
formulated the equations governing the energetics of the atmospheric general circulation.
He then simplified these equations to form a basis for numerical prediction of weather and
climate by computer models. His mathematical methodology has been widely used in making
numerical simulations of the atmospheric circulation in connection with global ecological
problems.
In the study of thermal convection in the gravitational field, he introduced the use of
low-order truncated representations of atmospheric dynamics to elucidate the fundamental
physics of atmospheric flow phenomena with minimal sets of nonlinear ordinary differential
equations. This approach led to his discovery of the first system of differential
equations, a dissipative system with a degree of freedom of only three, whose solution
constitutes the prototypical example of deterministic chaos. He discovered that
deterministic nonlinear dynamics can produce what as become known as the "butterfly
effect," in which a minute change in an initial state may well result in a huge
difference in a future state. Furthermore, by introducing the notions of transitivity and
intransitivity, he clarified the complexity of the climatic system and gave a theoretical
basis for abrupt climatic changes.
The "Lorenz Attractor" is a "simple" set of three deterministic
equations developed by Edward Lorenz while studying the non- repeatability of weather
patterns. The weather forecaster's basic problem is that even very tiny changes in initial
patterns ("the beating of a butterfly's wings" - the official term is
"sensitive dependence on initial conditions") eventually reduces the best
weather forecast to rubble. The lorenz attractor is the plot of the orbit of a dynamic
system consisting of three first order non-linear differential equations.
The solution to the differential equation is vector-valued function of one variable. If
you think of the variable as time, the solution traces an orbit. The orbit is made up of
two spirals at an angle to each other in three dimensions. We change the orbit color as
time goes on to add a little dazzle to the image.
The equations are:
You can solve these differential equations approximately using a method known as the first
order Taylor Series: by treating the notation for the derivative dx/dt as though it really
is a fraction, with "dx" the small change in x that happens when the time
changes "dt". Multiplying through the above equations by dt, yieldsl the change
in the orbit for a small time step.
A Strange Attractor
When you watch a storm long enough a pattern emerges.
When you give up hope of controlling it, you begin to see a beauty.
A man named Edward Lorenz found a butterfly in the math of a storm
a pattern that wrapped around a single point forever and unrepetitive
infinite in finite space.
He called it a strange attractor.
He tried to turn a storm into a mathematical model, and the storm
turned the equations into a butterfly and flew off.
It was wrapping around itself, over and over again
making the wings of a butterfly out of the equations of a storm.
The storm would never travel the same path around a point yet never
reach it.
I never thought someone could make a butterfly out of a hurricane
until
I met you.
You are the butterfly in the hurricane.
You are my strange attractor.
© Peter Moyes, 1997
Determinism is the belief that every action is the result of preceding actions. It
began as a philosophical belief in Ancient Greece thousands of years ago and was
introduced into science around 1500 A.D. with the idea that cause and effect rules govern
science. Sir Isaac Newton was closely associated with the establishment of determinism in
modern science. His laws were able to predict systems very accurately. They were
deterministic at their core because they implied that everything that would occur would be
based entirely on what happened right before. The Newtonian model of the universe is often
depicted as a billiard game in which the outcome unfolds mathematically from the initial
conditions in a pre-determined fashion, like a movie that can be run forwards or backwards
in time. Determinism remains as one of the more important concepts of physical science
today.
References:
Benoit Mandelbrot, (1977) The Fractal Geometry of Nature, New York: Freeman.
James Gleick, Chaos, New York: Penguin Books (1988) for a very readable
"popularized" social history of the emergence of chaos theory. Easy read
somewhat weak in terms of accuracy (non-rigorous treatment).
John Briggs and F. David Peat, Turbulent Mirror, New York: Harper and Row, (1988)
Arun Holden (1986) has a much more rigorous treatment but offers field research in
physics, physiology and other disciplines
Lorentz, E. N., 1963: "Deterministic Nonperiodic Flow", J. Atmos.
Sci. 20, pp. 130-141.
Palmer, T. N., 1993: "Extended-Range Atmospheric Prediction and the
Lorentz Model", Bulletin American Meteorological Society, pp. 49-65.
Pielke R. A. and Zeng, X., 1994: "Long-Term Variability of Climate", J. Atmos.
Sci. 51, pp.155-159.
Chaos and Philosophy
Chaos and Theology
Creating Chaos
Lorenz's Ants: Is Insect Behavior Chaotic?
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