Early chaos theory

Abstract

The article “Chaos at fifty” by Adilson Motter and David Campbell (Physics Today, May 2013, page 27) pays a well-deserved homage to Edward Lorenz for his contribution to chaos theory and meteorology. However, I take exception to the statement at the beginning of the article that “in 1963 an MIT meteorologist revealed deterministic predictability to be an illusion and gave birth to a field that still thrives.” On the contrary, the exponential growth of errors in some deterministic systems and the practical consequences for predictability have been appreciated by scientists for more than a century. I am particularly fond of what Henri Poincaré says in “Le hasard,” chapter 4 of his book Science et Méthode (Ernest Flammarion, 1908; my translation):A small cause, that escapes us, determines a considerable effect that we cannot ignore, and we then say that this effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial instant, we could predict exactly the situation of this same universe at a later instant.But Poincaré remarks that we know the initial situation only approximately, and that it may happen that small differences in initial conditions generate large differences later, so that prediction becomes impossible. Poincaré offers meteorology as an example:Why do meteorologists have such difficulty in predicting the weather with any certainty? … [They see] that a cyclone will appear, but they are unable to say where; a tenth of a degree added or subtracted at some arbitrary place, the cyclone appears here and not there, and causes destructions in countries which it would have spared. If one had been aware of this tenth of a degree, one could have known it in advance, but the observations were not spaced closely enough, and were not precise enough, and this is why everything seems due to chance. Here again we find the same contrast between a tiny cause, that the observer cannot measure, and considerable effects, which may be appalling disasters.So, Poincaré knew about the butterfly effect 50 years before Lorenz. Lorenz’s contribution is not so much the knowledge of the Lorenz attractor (a somewhat abstract model for atmospheric convection) as it is the demonstration that with computers, meteorologists can progressively improve their modeling of the dynamics of the atmosphere.Instead of a serendipitous discovery giving birth to a new field of science out of the blue, I see the blooming of chaos theory as a consequence of the progress in mathematical, experimental, and computational techniques, which over several decades have given rise to a formidable self-organized multidisciplinary effort.Using the mathematical theory of dynamical systems developed after Poincaré and Jacques Hadamard, and based on their work, Floris Takens and I, for instance, showed that Landau’s quasi-periodic theory of turbulence was unstable and led to hyperbolic dynamics and “strange attractors.”11. D. Ruelle, F. Takens, Comm. Math. Phys. 20, 167 (1971). https://doi.org/10.1007/BF01646553 That was an early contribution to what was not yet called chaos theory.A fundamental problem Poincaré explicitly left open was that of the stability of the solar system. The problem was not solved by the discovery of homoclinic tangles, because they may involve only sets of measure zero. The Kolmogorov-Arnold-Moser theory gave the hope that one could prove the stability of the solar system. But delicate computational work by Jack Wisdom and Jacques Laskar in the 1980s finally proved instability and thus solved the important classical problems of stability22. H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris (1899). (or long-term predictability). Laskar’s contributions in particular are chaos theory at its best:33. J. Laskar, Nature 338, 237 (1989). https://doi.org/10.1038/338237a0 They provide new views on the history of climates and other important geological questions.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. D. Ruelle, F. Takens, Comm. Math. Phys. 20, 167 (1971). https://doi.org/10.1007/BF01646553, Google ScholarCrossref2. H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris (1899). Google ScholarCrossref3. J. Laskar, Nature 338, 237 (1989). https://doi.org/10.1038/338237a0, Google ScholarCrossref© 2014 American Institute of Physics.