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

4. The impact of chaos on mathematics |

The mathematical studies of Smale [5] have shown that the orbit of a dynamical system is in some cases asymptotic to a complicated set called an Axiom A attractor. The behaviour of the system is then chaotic and, since Axiom A attractors can be analysed in great detail, they provide very important examples of chaos.

Another example of chaos was obtained in an early computer study by Lorenz [3] when he analysed a (rather brutally) simplified model of convection described by the following equations

_{}

_{}

_{}

with *s*=10, *b*=8/3, *r*=28. The Lorenz
attractor is chaotic, but different from the Axiom A attractors of Smale in that
it contains an (unstable) fixed point for the time evolution. Interestingly we
have, at this time, no mathematical proof that the solutions of the Lorenz
system behave (chaotically) as we think they do. We have however a model
inspired by the equations, and called geometric Lorenz attractor, for which a
detailed mathematical study has been given by Guckenheimer [6] and Williams [7].
It is known in particular that the *geometric Lorenz attractor* has some
properties of persistence when the equations are slightly changed (technically,
what is proved is co-dimension 2 structural
stability).