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If we replace "mathematics" with "nonlinear science," Stan's comment becomes even more appropriate to the present situation. We have already seen the beginnings of an understanding of many aspects of morphology, from fractal structures in ferns to nonlinear pattern-selection models for human digits. Similarly, the role of chaos in biological cycles, from heartbeats to cell densities, is rapidly being clarified. And the basic observation that incredibly complex behavior-including both pattern formation and self-reproduction can emerge in systems governed by very simple rules has obvious implications for modeling biological phenomena.
But the greatest challenge is clearly to understand adaptation, learning, and evolution. Adaptive complex systems will have features familiar from conventional dynamical systems, including hierarchical structures, multiple basins of attraction, and competition among many metastable configurations. In addition, they must also have a mechanism for responding to, and taking advantage of, changes in their environment. One approach to adaptation is to construct an explicit temporal hierarchy: one scale describes the actual dynamics and a second, slower time scale allows for changes in the nonlinear equations themselves. Models for the human immune system and for autocatalytic protein networks are among the prospective initial applications for this concept.
A second approach to adaptation, sometimes termed connectionism, is based on the idea that many simple structures exhibit complex collective behavior because of connections between the structures. Recent specific instances of this approach include mathematical models called neural networks. Although only loosely patterned after
Fig. 20. This cellular automaton consists of a grid of square cells with each cell able to take on any of eight possible states (indicated by different colors). (a) The basic building block of a repeating pattern for this automaton is a hollow square occupying an area of 10 by 15 cells with a tail that develops (b) until it produces a second hollow square. (c) The pattern continues to grow in time until (d) it has produced a large colony of the original pattern. (Figures courtesy of Chris Langton, Los Alamos National Laboratory.)
true neurological systems, such networks show remarkable promise of being able to learn from experience. A related set of adaptive models, called classifier systems, show an ability to self-generate a hierarchy of behavioral rules: that is, the hierarchy is not placed a priori into the system but develops naturally on the basis of the system's experience. In general, connectionist models suggest a resolution of the long-standing issue of building a reliable computer from unreliable parts.
In all these future developments, the tripartite methodology incorporating experimental mathematics, real experiments, and novel analytic approaches will continue to play a critical role. One very exciting prospect involves the use of ultraspeed interactive graphics, in which enormous data sets can be displayed visually and interactively at rates approaching the limits of human perception. By using color and temporal evolution, these techniques can reveal novel and unexpected phenomena in complicated systems.
To insure the long-term success of nonlinear science, it is crucial to train young researchers in the paradigms of nonlinearity. Also, interdisciplinary networks must be fostered that consist of scholars who are firmly based in individual disciplines but are aware of, and eager to understand, developments in other fields.
In all these respects, nonlinear science represents a singularly appropriate intellectual legacy for Stan Ulam: broadly interdisciplinary, intellectually unfettered and demanding, and very importantly-fun. ■
David Campbell, the Laboratory's first J. Robert Oppenheimer Fellow (from 1974 to 1977), is currently the Director of the Laboratory's Center for Nonlinear Science. He received his B.A. in chemistry and physics from Harvard College in 1966 and his Ph.D. in theoretical physics from Cambridge University in 1970. David has extended his activities in physics and nonlinear science to the international level, having been a National Academy of Sciences Exchange Scientist to the Soviet Union in 1977, a Visiting Professor at the University of Dijon, Dijon, France in 1984 and 1985, and a Ministry of Education Exchange Scientist to the People's Republic of China in 1986. He and his wife, Ulrike, have two children, Jean-Pierre and Michael.
I am grateful to the many colleagues who, over the years, have shared their insights and helped shape my perspective on nonlinear science. In the preparation of this article I have benefited greatly from the comments and assistance of Jim Crutchfield, Roger Eckhardt, Doyne Farmer, Mitchell Feigenbaum, Jim Glimm, Erica Jen, and Gottfried Mayer-Kress. I wish also to thank my generous coworkers in nonlinear science who permitted the use of the figures and pictures so essential to this article.
General Discussions of Nonlinear Science and Background
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Solitons and Coherent Structures
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Deterministic Chaos and Fractals
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G. Mayer-Kress and H. Haken. 1981. The influence of noise on the logistic model. Journal of Statistical Physics 26:149–171.
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Complex Configurations and Pattern Formation
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Adaptive Nonlinear Systems
Doyne Farmer, Alan Lapedes, Norman Packard, and Burton Wendroff, editors. 1986. Evolution, games, and learning: models for adaptation in machines and nature: Proceedings of the Fifth Annual International Conference of the Center for Nonlinear Studies, Los Alamos. Amsterdam: North-Holland Publishing Co. (reprinted from Physica D 22:Nos. 1-3).
THE ERGODIC HYPOTHESIS
by Adrian Patrascioiu
here are a few problems in physics that stir deep emotions every time they are discussed. Since physicists are not generally speaking an emotional group of people, the existence of these sensitive issues must be considered a strong indication that something is amiss. One such issue is the interpretation of quantum mechanics. I will take a moment to discuss that problem because it bears directly on the main topic of this article.
In quantum mechanics, if the question asked is a technical one, say how to compute the energy spectrum of a given atom or molecule, there is universal agreement among physicists even though the problem may be analytically intractable. If on the other hand the question asked pertains to the theory of measurement in quantum mechanics, that is, the interpretation of certain experimental observations performed on a microscopic system, it is virtually impossible to find two physicists who agree. What is even more interesting is that usually these controversies are void of any physical predictions and are entirely of an epistemological character. They reflect our difficulty in bridging the gap between the quantum mechanical treatment of the microscopic system being observed and the classical treatment of the macroscopic apparatus with which the measurement is performed. It is usually argued that we, physicists, have difficulty comprehending the formalism of quantum mechanics because our intuition is macroscopic, hence classical, in nature. Now if that were the case, we should have as much difficulty with special relativity, since we are hardly used to speeds comparable to that of light. Yet, strange as it seems at first, I have never heard physicists argue about the "twin paradox," the