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Polytechnic School's 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 two-body solution gives analytic confirmation of the great Johannes Kepler's empirically derived laws of planetary motion. Poincare found that the numerical "solution" of the three-body 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 three-body 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 near-perfect predictability. Yet Poincaré's model for three bodies is just an extension of Newton's two-body 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 two-body 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 non-periodic 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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