
Mathematical Modeling II
with Calculus BC
Chaos Theory
Chaos theory is very much a 20th century
development, but the man who probably best deserves the title "Father of Chaos Theory"
was a great French mathematician of the 19th century named Henri Poincaré. As
discussed on the Dynamical Systems page, Isaac Newton had
given the world what seemed to be the final word on how the solar system worked. But
Poincaré made the observation that Newton's beautiful model was posited on the basis
of the interaction between just two bodies. That is all Newton's differential equations
allow. It was natural for anyone with an inquiring mind to wonder what would
happen if three or more bodies were allowed in the model. In fact, the question became
so famous that a prize was offered for its solution and it was given a name  "The
Three Body Problem." Poincaré, being one of the preemminent mathematicians of the
time, tried his hand at solving the Three Body Problem. Ironically, he ended up
winning the prize by writing a paper showing that he could not solve it. The problem
was that, while Newton's differential equations for two bodies have nice clean "closed
form" solutions, the equations for three bodies do not. They must be "solved" by
approximate numerical techniques, which effectively change the modeling process from
continuous to discrete. The twobody solution gives analytic confirmation of the great
Johannes Kepler's empirically derived laws of planetary motion. Poincare found that
the numerical "solution" of the threebody problem revealed orbits "so tangled that I
cannot even begin to draw them." In addition, Poincaré discovered a very disturbing
fact: when the three bodies were started from slightly different initial positions, the
orbits would trace out drastically different paths. He wrote, "It may happen that small
differences in the initial positions may lead to enormous differences in the final
phenomena. Prediction becomes impossible." This is the statement which gives Poincare
the claim to the title "Father of Chaos Theory." This is the first known published
statement of the property now known as "sensitivity to initial conditions", which is one
of the defining properties of a chaotic dynamical system.
Poincaré's conclusions about the threebody problem were undeniably correct, but
also unwelcome in Newton's perfectly ordered universe. Science is very much about
making predictions of future events based upon laws derived from the observation of past
events. Deterministic models, so it was thought, must yield perfect or nearperfect
predictability. Yet Poincaré's model for three bodies is just an extension of Newton's twobody
model and is, therefore, also deterministic. But Poincare says that in the context of
this model "prediction becomes impossible" in some instances. From the time of Newton
until the time of Poincaré, scientists had experienced too many spectacular successes
to simply jettison the comfortable clockwork predictability of Newton's twobody calculus
in favor of Poincaré's disturbing uncertainties. Besides they had no palatable way to
deal with the numerical drudgery involved in calculating Poincaré's discretely computed
orbits. Thus Poincare's monumental discovery of what we now call deterministic chaos was
destined to be conveniently placed on the scientific back burner for the next seven
decades before being rediscovered by a mathematically trained meteorolgist at the
Massachusetts Institute of Technology.
Weather has always been a problem for mainstream science, because every effort to
predict it has produced results that range from mixed to horrible. So in the 1950s when
digital computers began to come on line in increasing numbers, meteorologists were optimistic
that at last they would be able to constuct models sophisticated enough to take into account
the many variables involved in weather forecasting. By the beginning of the sixties, however,
despite the employment of elaborate computer models, weather prediction showed little,
if any, improvement from a decade earlier. In the early sixties, Ed Lorenz, a meteorolgist
who had studied mathematics at Harvard, was trying out the new computer that had recently
been delivered to his office at the Massachusetts Institute of Technology when he stumbled
upon an oddity in the weather model he was testing on the computer. The model consisted
of a typically deterministic system of differential equations, inherently no more complicated
than Newton's laws of motion. Lorenz wanted to take a second look at several months of
simulated weather generated by the computer. So he stopped the simulation, reentered the
initial values for his variables and started the simulation over from the beginning. Of
course, since this was a completely deterministic model, he expected to see a pattern of
values identical to the previous simulation. He was puzzled and somewhat irritated to
find that after a month or two of simulated weather the values being plotted on the screen
were completely different from those of the first simulation. After testing for computer
malfunction and rerunning the program several times, he finally discovered the problem.
In entering the initial values, he had used three decimal places of accuracy, while the
computer was internally working with six places. Lorenz had rediscovered Poincaré's
"sensitivity to initial conditions." Lorenz called it "the Butterfly Effect", because it
implied that if a butterfly in Brazil flapped its wings it could set off a chain reaction
in weather patterns that might lead to a tornado in Texas a few weeks later. In his 1963
paper for a meterological journal, he called this behavior "deterministic nonperiodic
flow." Some thirteen years later, Maryland physicist James Yorke would give such phenomena
the name that, for better or worse, would stick: "chaos." In the MMII/BC course, you will
learn about the scientific and mathematical work carried out between Lorenz's accidental discovery and
Yorke's christening of the new science. You will also learn of the many new discoveries
in chaos theory that have come in the quarter century since Yorke gave the discipline its
name. For the most complete and readable history of the development of chaos theory, read
James Gleick's thoroughly engrossing Chaos:Making a New
Science.
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