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

4. The impact of chaos on mathematics |

The phenomenon called turbulence is complex and difficult to analyse. What we know about it results from experimental, numerical, and mathematical studies. One important feature of turbulence is that, as claimed by Ruelle and Takens [4], it is chaotic: this has been proved experimentally by the observation of strange attractors, Feigenbaum bifurcations, etc. at the onset of turbulence. A mathematical proof of chaos in the Navier - Stokes equation does not exist at this time. (And when one is obtained, it will no longer create much excitement.) We have thus here a very interesting situation from the epistemological viewpoint, where we are firmly convinced of a certain mathematical fact (the existence of chaos in the solutions of the Navier - Stokes equations) but our belief is based on experimental evidence.

Another conceptually important question arises in the theory of
turbulence in relation with chaos: this is the question of *natural
measures* or *SRB measures*. Consider a hydrodynamical system kept in
turbulent motion by the action of some time-independent external forces. One
expects that time averages can be defined for the turbulent motion. Time
averages correspond to "ensemble averages", i.e., to a probability measure
invariant by time evolution (in fact ergodic). The relevant ergodic measure has
support on the attractor describing the motion. In the framework of the Hopf -
Landau theory of turbulence, the attractor is quasiperiodic and carries only one
invariant measure. But a strange attractor carries uncountably many distinct
ergodic measures! Which one should we use to represent time averages? As it
happens we have a reasonable answer to this question for certain dynamical
systems in terms of so-called natural or SRB measures (see next section). But
independently of the answer, it is conceptually important to recognize that
there is a problem. For many years, people have tried to determine time averages
by "solving the Hopf equation" (this equation only expresses invariance of a
probability measure under the Navier - Stokes time evolution). It is now clear
that an essential ingredient beyond the Hopf equation is needed to determine
time averages. (Ad hoc truncations of the equations, and assumptions of
gaussianity are unlikely to be the answer, see Ruelle
[13]).