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 CELLULAR AUTOMATON 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. 1 Acknowledgments 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. Further Reading General Discussions of Nonlinear Science and Background H. H. Goldstein and J. von Neumann. 1961. On the principles of large scale computing machines. In John von Neumann: Collected Works, Volume V, edited by A. H. Taub, pp. 1–32. New York: Pergamon Press. E. Fermi, J. Pasta, and S. Ulam. 1965. Studies of nonlinear problems. In Enrico Fermi: Collected Papers, Volume II, pp. 978-988. Chicago: University of Chicago Press. David Campbell, Jim Crutchfield, Doyne Farmer, and Erica Jen. 1985. Experimental mathematics: the role of computation in nonlinear science. Communications of the ACM 28:374–384. Solitons and Coherent Structures D. J. Korteweg and G. DeVries. 1895. On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philosophical Magazine 39:422-443. Alfred Segger, Hans Donth, and Albert Kochendörfer. 1953. Theorie der Versetzungen in eindimensionalen Atomreihen. III. Versetzungen, Eigenbewegungen und ihre Weshselwirkung. Zeitschrift für Physik. 134:171– 193. J. K. Perring and T. H. R. Skryme. 1962. A model unified field equation. Nuclear Physics 31:550-555. N. Zabusky and M. Kruskal. 1965. Interaction of "solitons" in a collisionless plasma and the recurrence of initial states. Physical Review Letters 15:240–243. Alwyn C. Scott, F. Y. F. Chu, and David W. McLaughlin. 1973. The soliton: a new concept in applied science. Proceedings of the IEEE 61:1443-1483. A. R. Osborne and T. L. Burch. 1980. Internal solitons in the Andaman Sea. Science 208:451-460. C. H. Tze. 1982. Among the first texts to explain the "soliton revolution." Physics Today, June 1982, 55-56. (This article is a review of Elements of Soliton Theory by G. L. Lamb, Jr., Solitons: Mathematical Methods for Physicists by G. Eilenberger, and Solitons and the Inverse Scattering Transform by M. Ablowitz and H. Segur.) Akira Hasegawa. 1984. Numerical study of optical soliton transmission amplified periodically by the stimulated Raman process. Applied Optics 23:3302-3309. David K. Campbell, Alan C. Newell, Robert J. Schrieffer, and Harvey Segur, editors. 1986. Solitons and Coherent Structures: Proceedings of the Conference on Solitons and Coherent Structures held at Santa Barbara. Amsterdam: North-Holland Publishing Co. (reprinted from Physica D 18:Nos. 1–3). C. G. Slough, W. W. McNairy, R. V. Coleman, B. Drake, and P. K. Hansma. 1986. Charge-density waves studied with the use of a scanning tunneling microscope. Physical Review B 34:994–1005. L. F. Mollenauer, J. P. Gordon, and M. N. Islam. 1986. Soliton propagation in long fibers with periodically compensated loss. IEEE Journal of Quantum Electronics QE-22:157–173. Deterministic Chaos and Fractals Henri Poincaré. 1952. Science and Method, translated by Francis Maitland. New York: Dover Publications, Inc. W. A. Bentley and W. J. Humphreys. 1962. Snow Crystals. New York: Dover Publications, Inc. M. V. Berry. 1978. Regular and irregular motion. In Topics in Nonlinear Dynamics: A Tribute to Sir Edward Bullard, edited by S. Jorna, A.I.P. Conference Proceedings, No. 46, pp. 16-120. New York: American Institute of Physics. Edward N. Lorenz. 1979. On the prevalence of aperiodicity in simple systems. In Global Analysis: Proceedings of the Biennial Seminar of the Canadian Mathematical Congress, edited by M. Grmela and J. E. Marsden, pp. 53-75. New York: Springer Verlag. Mitchell J. Feigenbaum. 1980. Universal behavior in nonlinear systems. Los Alamos Science 1 (Summer 1980):4-27 (reprinted in Physica D 7:16–39, 1983). Robert H. G. Helleman. 1980. Self-generated chaotic behavior in nonlinear mechanics. In Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen, pp. 165-233. Amsterdam: North-Holland Publishing Co. B. A. Huberman, J. P. Crutchfield, and N. H. Packard. 1980. Noise phenomena in Josephson junctions. Applied Physics Letters 37:750-753. J.-P. Eckmann. 1981. Roads to turbulence in dissipative dynamical systems. Reviews of Modern Physics 53:643-654. G. Mayer-Kress and H. Haken. 1981. The influence of noise on the logistic model. Journal of Statistical Physics 26:149–171. Benoit Mandelbrot. 1983. The Fractal Geometry of Nature. New York: W. H. Freeman and Company. David K. Umberger and J. Doyne Farmer. 1985. Fat fractals on the energy surface. Physical Review Letters 55:661-664. Gérard Daccord, Johann Nittmann, and H. Eugene Stanley. 1986. Radial viscous fingers and diffusion-limited aggregation: fractal dimension and growth sites. Physical Review Letters 56:336–339. Johann Nittmann and H. Eugene Stanley. 1986. Tip splitting without interfacial tension and dendritic growth patterns arising from molecular anisotropy. Nature 321:663–668. James Gleick. 1987. Chaos: Making a New Science. New York: Viking Penquin, Inc. David K. Campbell. 1987. Chaos: chto delat? To be published in Nuclear Physics B, Proceedings of Chaos, 1987. Complex Configurations and Pattern Formation J. D. Farmer, T. Toffoli, and S. Wolfram, editors. 1984. Cellular Automata: Proceedings of an Interdisciplinary Workshop, Los Alamos. Amsterdam: North-Holland Publishing Co. (reprinted from Physica D 10:Nos. 1-2). Alan R. Bishop, Laurence J. Campbell, and Paul J. Channell, editors. 1984. Fronts, Interfaces, and Patterns: Proceedings of the Third Annual International Conference of the Center for Nonlinear Studies, Los Alamos. Amsterdam: North-Holland Publishing Co. (reprinted from Physica D 12:1–436). Basil Nicolaenko (Nichols). 1987. Large scale spatial structures in two-dimensional turbulent flows. To be published in Nuclear Physics B, Proceedings of Chaos, 1987. Pierre Bergé. 1987. From temporal chaos towards spatial effects. To be published in Nuclear Physics B, Proceedings of Chaos, 1987. Paul R. Woodward, David H. Porter, Marc Ondrechen, Jeffrey Pedelty, Karl-Heinz Winkler, Jay W. Chalmers, Stephen W. Hodson, and Norman J. Zabusky. 1987. Simulations of unstable flow using the piecewiseparabolic method (PPM). In Science and Engineering on Cray Supercomputers: Proceedings of the Third International Symposium, pp. 557–585. Minneapolis: Cray Research, Inc. Alan C. Newell. The dynamics of patterns: a survey. In Propagation in Nonequilibrium Systems, edited by J. E. Wesfried. New York: Springer Verlag. To be published. 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 A Complicated and Physics by Adrian Patrascioiu T 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 |