Complexity_ A Guided Tour - Melanie Mitchell [149]
“The values of x … become chaotic”: How do we know that the system will not settle down to a regular oscillation after a large number of iterations? This can be shown mathematically; e.g., see Strogtaz, S., Nonlinear Dynamics and Chaos. Reading, MA: Addison-Wesley, 1994, pp. 368–369.
“a basis for constructing pseudo-random numbers”: A pseudo-random number generator is a deterministic function or algorithm that outputs values whose distribution and lack of correlation satisfies certain statistical tests of randomness. Such algorithms are used in modern computing for many different applications. Using the logistic map as a basis for pseudo-random number generators was first suggested in Ulam, S. M. and von Neumann, J., On combination of stochastic and deterministic processes (abstract). Bulletin of the American Mathematical Society, 53, 1947, p. 1120. This has been further investigated by many others, for example, Wagner, N. R., The logistic equation in random number generation. In Proceedings of the Thirtieth Annual Allerton Conference on Communications, Control, and Computing, University of Illinois at Urbana-Champaign, 1993, pp. 922–931.
“The fact that the simple and deterministic equation”: May, R. M., Simple mathematical models with very complicated dynamics. Nature, 261, 1976, pp. 459–467.
“The term chaos … T. Y. Li and James York”: Li, T. Y. and Yorke, J. A., Period three implies chaos. American Mathematical Monthly 82, 1975, p. 985.
“The period-doubling route to chaos has a rich history”: For an interesting history of the study of chaos, see Aubin, D. and Dalmedico, A. D., Writing the history of dynamical systems and chaos: Longue Durée and revolution, disciplines, and cultures. Historia Mathematica 29, 2002, pp. 273–339.
“Feigenbaum adapted it for dynamical systems theory”: For an accessible explanation of Feigenbaum’s theory, see Hofstadter, D. R., Mathematical chaos and strange attractors. In Metamagical Themas. New York: Basic Books, 1985.
“the best thing that can happen to a scientist”: Quoted in Gleick, J., Chaos: Making a New Science. New York: Viking, 1987, p. 189.
“certain computer models of weather”: see, e.g., Selvam, A. M.. The dynamics of deterministic chaos in numerical weather prediction models. Proceedings of the American Meteorological Society, 8th Conference on Numerical Weather Prediction, Baltimore, MD, 1988; and Lee, B. and Ajjarapu, V., Period-doubling route to chaos in an electrical power system. IEE Proceedings, Part C, 140, 1993, pp. 490–496.
“Pierre Coullet and Charles Tresser, who also used the technique of renormalization”: Coullet, P., and Tresser, C., Itérations d’endomorphismes et groupe de renormalization. Comptes Rendues de Académie des Sciences, Paris A, 287, 1978, pp. 577–580.
Chapter 3
“The law that entropy increases”: Eddington, A. E., The Nature of the Physical World. Macmillan, New York, 1928, p. 74.
“complex systems sense, store, and deploy more information”: Cohen, I., Informational landscapes in art, science, and evolution. Bulletin of Mathematical Biology, 68, 2006, p. 1218.
“evolution can perform its tricks”: Beinhocker, E. D., The Origin of Wealth: Evolution, Complexity, and the Radical Remaking of Economics. Cambridge, MA: Harvard Business School Press, 2006, p. 12.
“Although they differ widely in their physical attributes”: Gell-Mann, M., The Quark and the Jaguar. New York: W. H. Freeman, 1994, p. 21.
“Why the second law should”: Rothman, T., The evolution of entropy. In Science à la Mode. Princeton, NJ: Princeton University Press, 1989, p. 82.
“the hot system”: Maxwell, quoted in Leff, H. S. and Rex, A. F., Maxwell’s Demon: Entropy, Information, Computing. Princeton University Press. Second edition 2003, Institute of Physics Pub., 1990, p. 5.
“In a famous paper”: Szilard, L., On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings.