Early chaos theory

Abstract

Motter and Campbell reply: We thank Professors David Ruelle and Dima Shepelyansky for their clarifying comments, which expand on some important aspects of the rich history of chaos that the stringent length and number of reference limits of Physics Today did not allow us to include in our article. We chose to focus our article on the contributions of Edward Lorenz and the role of computation in the development of the modern theory of chaos.We are well aware of, and in our article we explicitly quoted from, Henri Poincaré’s insights into “sensitive dependence on initial conditions.” Indeed, almost precisely the same paragraphs that Ruelle quotes in his letter appeared in an article by one of us published more than 25 years ago.11. D. K. Campbell, Los Alamos Sci. 15, 218 (1987) http://library.lanl.gov/cgi-bin/getfile?15-17.pdf; reprinted in From Cardinals to Chaos: Reflections on the Life and Legacy of Stanislaw Ulam, N. G. Cooper, ed.,Cambridge U. Press, New York (1989), p. 218. Had space permitted, we would also have included quotes from James Maxwell,22. B. R. Hunt, J. A. Yorke, Nonlinear Sci. Today 3 (1), 1 (1993).,33. J. C. Maxwell, “Essay for the Eranus Club on Science and Free Will, February 11, 1873,” in L. Campbell, W. Garnett, The Life of James Clerk Maxwell, MacMillan and Co, London (1882), p. 434. who, decades before Poincaré, clearly recognized that sensitive dependence on initial conditions implies loss of predictability. As noted by Richard Kautz,44. R. Kautz, Chaos: The Science of Predictable Random Motion, Oxford U. Press, New York (2011), p. 165. “it is perhaps fairest to say that chaos was discovered many times, although most discoverers did not understand their discovery as fully as Lorenz.”Our focus on Lorenz’s work was also motivated by its central role in bringing the quantitative aspects of chaos to the awareness of the scientific community. This is reflected in the paper that named the field,55. T.-Y. Li, J. A. Yorke, Am. Math. Mon. 82, 985 (1975). https://doi.org/10.2307/2318254 in which the first four references were to publications by Lorenz.We are pleased that Ruelle’s final comments on the importance of Jack Wisdom and Jacques Laskar’s “delicate computational work” reinforce our point about the essential role played by computation—both the numerical results and the visualizations—in the full development of chaos theory and its applications. That point is discussed in detail in reference 12 of our Physics Today article.Shepelyansky’s remarks about the significance of the work of his mentor and close collaborator Boris Chirikov in developing an approximate theoretical approach—the Chirikov overlap criterion—to the study of chaos in Hamiltonian systems are pertinent. We chose to focus our brief discussion of Hamiltonian chaos on the more general and prior Kolmogorov-Arnold-Moser theory,66. V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer-Verlag, New York (1989). mentioned in Ruelle’s letter. Interested readers are encouraged to consult Chirikov’s papers. As noted at the end of our article, “There have been many other important developments in chaos that could not be discussed in this brief, nontechnical article.”REFERENCESSection:ChooseTop of pageREFERENCES <<1. D. K. Campbell, Los Alamos Sci. 15, 218 (1987) http://library.lanl.gov/cgi-bin/getfile?15-17.pdf; Google Scholarreprinted in From Cardinals to Chaos: Reflections on the Life and Legacy of Stanislaw Ulam, N. G. Cooper, ed.,Cambridge U. Press, New York (1989), p. 218. Google Scholar2. B. R. Hunt, J. A. Yorke, Nonlinear Sci. Today 3 (1), 1 (1993). Google ScholarCrossref3. J. C. Maxwell, “Essay for the Eranus Club on Science and Free Will, February 11, 1873,” in L. Campbell, W. Garnett, The Life of James Clerk Maxwell, MacMillan and Co, London (1882), p. 434. Google Scholar4. R. Kautz, Chaos: The Science of Predictable Random Motion, Oxford U. Press, New York (2011), p. 165. Google Scholar5. T.-Y. Li, J. A. Yorke, Am. Math. Mon. 82, 985 (1975). https://doi.org/10.2307/2318254, Google ScholarCrossref, ISI6. V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer-Verlag, New York (1989). Google ScholarCrossref© 2014 American Institute of Physics.