people into friends and strangers.'' Linear systems are friends, "occasionally quirky [but] essentially understandable." Nonlinear systems are strangers, presenting "quite a different problem. They are strange and mysterious, and there is no reliable technique for dealing with them" (Hubbard and West, 1991, p. 1).

In particular, many of these nonlinear systems have been found to exhibit a surprising mix of order and disorder. In the 1970s, mathematician James Yorke from the University of Maryland's Institute for Physical Science and Technology used the word chaos to describe the apparently random behavior of a certain class of nonlinear systems (York and Li, 1975). The word can be misleading: it refers not to a complete lack of order but to apparently random behavior with nonetheless decipherable pattern. But the colorful term captured the popular imagination, and today chaos is often used to refer to the entire nonlinear realm, where randomness and order commingle.

Now that scientists are sensitive to the existence of chaos, they see it everywhere, from strictly mathematical creations to the turbulence in a running brook and the irregular motions of celestial bodies. Nonrigorous but compelling evidence of chaotic behavior in a wide array of natural systems has been cited: among others, the waiting state of neuronal firing in the brain, epidemiologic patterns reflecting the spread of disease, the pulsations of certain stars, and the seasonal flux of animal populations. Chaotic behavior has even been cited in economics and the sociology of political behavior.

To what extent the recognition of chaos in such different fields will lead to scientific insights or real-world applications is an open question, but many believe it to be a fundamental tool for the emerging science of complexity.

Certainly the discovery of "chaos" in dynamical systems—and the development of mathematical tools for exploring it—is forcing physicists, engineers, chemists, and others studying nonlinear processes to rethink their approach. In the past, they tended to use linear approximations and hope for the best; over the past two or three decades they have come to see that this hope was misplaced. It is now understood that it is simply wrong to think that an approximate model of such a system will tell you more or less what is going on, or more or less what is about to happen: a slight change or perturbation in such a system may land you somewhere else entirely.

Such behavior strikes at the central premise of determinism, that given knowledge of the present state of a system, it is possible to project its past or future. In his presentation Hubbard described how the mathematics underlying dynamical systems relegates this deterministic view of the world to a limited domain, by destroying confi-



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