by Ralph H. Abraham

Q1: What is dynamical systems theory?

Dynamical systems theory is the branch of mathematics devoted to the motions of systems which evolve according to simple rules. It was developed originally in the 17th century by Newton to model the motions of our solar system, evolving under the rules of his new theory of universal gravitation.

Q2: What is chaos theory?

Chaos theory is a further development of dynamical systems theory which focusses on highly complex motions called chaotic motions. These were discovered originally by Poincare around 1890 in his unsuccesful efforts to prove the stability of the solar system.

Q3: What is a chaotic motion?

By the broadest definition, every motion more complicated than fixed (no motion) or periodic (cyclicly repeating) motion is considered chaotic.

Q4: What do you mean by the chaos revolution?

Chaos theory was not known by this name until 1975 or so. For almost a century it was a minor undercurrent of dynamical systems theory, a topic of pure mathematics unknown to the scientfic community. After the computer revolution, chaotic motions became visible on computer graphic screens, and an awareness of their significance began to spread among scientists. In 1971 this awareness materialized as a technical report published in a journal of theoretical physics, and the chaos revolution was on. It took about 15 years more to sweep throughout the sciences and reach public awareness. All this aounts to a major paradigm shift, as chaotic behavior moved from mystery to familiarity.

Q5: Why is chaos theory important?

Every branch of pure mathematics has applications, usually to science or technology, which are important to society. In the case of dynamical systems theory, extensive and ongoing applications to all of the physical, biological, and social sciences have been fundamental to our evolving culure. The most frequent kind of application is to the technology of modeling complex natural systems. The importance of chaos theory has been in the context of this modeling aspect of applied dynamics. Because of the new wisdom of chaotic motions, many more complex systems now have useful models: the biosphere, the global economy, the human immune system, and so on. Different models for subsystems, created by scientists of disjoint specialities, may now be combined into a single complex supemodel, thanks to chaos theory. It provides a new technique for the unification of the sciences.

Q6: Is the chaos of chaos theory the same as the chaos in everyday life?

Off hand it is not obvious that the chaotic motions of chaos have any direct bearing on the chaotic experiences of everyday life. However, as the applications of chaos theory to the social sciences evolve, more and more everyday chaos is brought into the embrace of chaos theory.

Q7: Is chaos theory taught in universities?

Sadly, no. While a small number ofuniversities offer a course in chaos theory in their physics or math departments, there are compreensive chaos theory programs taught be experts in the field only at a handfull of universities.

Q8: What background is required to understand chaos theory?

As chaos theory is a new branch of math, it is relatively independent of the main topics of the trditional progrtam. Therefore it is quitte accessible to people without extensive math background. Calculus, for example, is not required. The plane geometry of Euclid is an excellent preparation.

Q9: Should chaos theory be taught in high schools?

Absolutely. The applicability of chaos theory to the complex space-time patterns observed in nature and in human society make it an important subject for everyone to learn. And the paucity of background knowledge required makes it accessible to all.

Q10: How does chaos theory relate to math anxiety?

Math anxiety is usually trigerred by high school algebra and its arcane symbolic notations, while most students feel confortable with geometry. As chaos theory builds upon geometry without requiring algebra, it provides a fresh start for those afflicted with math anxiety, and may actually restore math confidence.
Revised 22 July 1999 by Ralph