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close this bookThe Impact of Chaos on Science and Society (UNU, 1997, 415 pages)
close this folder4. The impact of chaos on mathematics
View the document(introduction...)
View the documentAbstract
View the documentI. Introduction: A historical view
View the documentII. The Lorenz attractor
View the documentIII. The Feigenbaum bifurcation
View the documentIV. Hydrodynamic turbulence
View the documentV. Ergodic theory of differentiable dynamical systems: Axiom A systems
View the documentVI. Ergodic theory of differentiable dynamical systems: General systems
View the documentVII. Quadratic maps of the interval and the Hénon attractor
View the documentVIII. Zeta functions
View the documentIX. Conclusion
View the documentReferences

Abstract

The relations of chaos and mathematics have been complex and subtle. Actually, chaos was discovered at the end of the nineteenth century by the mathematicians J. Hadamard and H. Poincaré, who largely appreciated the philosophical implications of their discovery. It was not possible, however, to deal with the subject in a quantitative fashion at that time. The rediscovery of chaos in recent decades has led to the sort of quantitative analysis that is characteristic of hard science, with important repercussions for pure mathematics. We review a number of topics that show the great diversity of the relations between chaos and mathematics: the Lorenz attractor, the Feigenbaum bifurcation, hydrodynamic turbulence, the ergodic theory of differentiable dynamical systems (Axiom A or more general systems), quadratic maps and the Hénon attractor, dynamical zeta functions. This is not an exhaustive list but it shows how the interplay of ideas coming from several parts of physics (statistical mechanics, hydrodynamics, etc.) and of mathematics has led to remarkable new insights, and technical progress both in physics and in mathematics.