History of Chaos Theory
Chaos theory was formulated during the 1960s. Its story is one of many
people--scientists who dared to think along new and unsuspected channels. The
name chaos was coined by Jim Yorke, an
applied mathematician at the University of Maryland (Ruelle, 1991).
In 1961, Edward Lorentz discovered the butterfly
effect while trying to forecast the weather. He was running a long series of
computations on a computer when he decided he needed another run. Rather than do
the entire run again, he decided to save some time by typing in some numbers
from a previous run. Later, when he looked over the printout, he found an
entirely new set of results. The results should have been the same as before. After thinking about this unexpected result, he discovered
that the numbers he typed in had been slightly rounded off.
In principle, this tiny difference in initial conditions should not have
made any difference in the result, but it did. From this, Lorentz determined
that long-distant weather forecasts are impossible to predict. Tiny differences
in weather conditions, on any one day, will show dramatic differences, after a
few weeks, and these differences are entirely unpredictable. Although Lorentz's
discovery was an accident, it planted the seed for the new theory of chaos
Mathematicians have known about nonlinearity
(a characteristic of discontinuous events) since, the work of
Henri Poincaré, at the turn of this century. Nonlinear equations have
been around for a long time, but no one was able to solve them, and traditional
scientists and engineers simply ignored all nonlinear portions of their
calculations. Most equations that attempt to predict the actions of nature or
natural materials are close approximations rather than exact. They contain one
or more factors of nonlinearity; which are approximated by using constants (in
the engineering community, such constants are sometimes called fudge
factors, we refer to them, in this book, as Chaos
Several courageous scientists were so intrigued with the new concept of
chaos, that they began to do research on both nonlinearity and turbulence
(any condition where orderly motion is broken up into random or chaotic motion).
However, according to Gleick (1987), they were warned by their supervisors and
colleagues that such research could cost them their respectability, and possibly
At that time, chaos was not a science, or even a cohesive theory, but
rather, an untested discipline with no real experts. Early researchers, in this
area, worked long and hard to develop their thoughts and findings into
publishable and acceptable forms. New terms were needed. Above all, a new way of
looking at the universe was required.
Traditional scientists were hardly aware of this emerging science until
most of the details had been worked out. Even then, some were strongly opposed
to it. But, with the help of the home computer, chaos science grew until, today,
it is an accepted scientific discipline in its own right.
What Lorentz did for weather, Robert May did for ecology. His work, in
the early 1970s, helped pin down the concepts of bifurcation
(a split or division by two) and period
doubling (one way in which order breaks down into chaos). A period is the
time required for any cyclic system to return to its original state. In the
early 1970s, May was working on a model that addressed how insect birthrate
varied with food supply. He found that at certain critical values, his equation
required twice the time to return to its original state--the period having
doubled in value. After several period-doubling cycles, his model became
unpredictable, rather like actual insect populations tend to be unpredictable.
Since May’s discovery with insects, mathematicians have found that this
period-doubling is a natural route to chaos for many different systems.
One of the foremost contributors to the new science was Benoit Mandelbrot.
Using a home computer, Mandelbrot (1982) pioneered the mathematics of fractals,
a term which he coined in 1975. His fractals (the geometry of fractional
dimensions) helped describe or picture the actions of chaos, rather than explain
it. Chaos and its workings could now be seen in color on a home computer.
The striking principle he discovered was that many of the irregular
shapes that make up the natural world, although seemingly random and chaotic in
form, have a simple organizing principle (Stwertka, 1987, p. 73). A new geometry
of chaos was born.
In 1971, David Ruelle and Floris Takens described a phenomena they called
a strange attractor (a special type of
attractor today called a chaotic attractor).
This strange phenomena was said to reside in what they called phase space (a geometric depiction of the state space of a system)
and a whole new element of chaos theory was born.
Phase space allows scientists to map information from complex systems,
make a picture of their moving parts, and allows insight into a dynamic system's
possibilities. It is a mathematically constructed conceptual space where each
dimension corresponds to one variable of the system (Kellert, 1993).
According to Ruelle (1991), his association of turbulence with a strange
attractor was so revolutionary, he was not able to publish his paper, and
finally published it himself. He writes, “Actually, I was an editor of the
journal, and I accepted the paper for publication.
This is not a recommended procedure in general, but I felt that it was
justified in this particular case” (p. 63).
Another pioneer of the new science was Mitchell Feigenbaum. His work, in
the late 1970s, was so revolutionary that several of his first manuscripts were
rejected for publication because they were so novel they were considered
irreverent (Gleick, 1987). He
discovered order in disorder. He looked deeply into turbulence, the home of
strange attractors, and saw universality. He developed a method to measure
turbulence and found a structure embedded in nonlinear systems. According to
Gleick (1987), a mathematical proof of his ideas was presented in 1979 by Oscar
E. Lawford III.
Feigenbaum showed that period doubling is the normal way that order
breaks down into chaos. He calculated universal numbers which represent ratios
in the scale of transition points that occur during the process of period
doubling. These ratios are now called Feigenbaum numbers. Gleick (1987) mentions
that Richard J. Cohen, and his medical colleagues at MIT, found that period
doubling is associated with the onset of a heart attack. This finding brought
chaos science into the domain of medical science.
By the mid 1970s, the movement toward chaos as a science was well underway, and in 1977, the first conference on chaos theory was held in Italy. Perhaps the most startling finding to come out of this new scientific theory is that order exists within chaos. In fact, order comes from chaotic conditions.