The Impact of Chaos on Science and Society (UNU, 1997, 415 pages) |

4. The impact of chaos on mathematics |

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.