Strange attractors and turbulence

The way in which probabilistic considerations play a natural role in the description of long-term behavior of solutions of deterministic differential equations is demonstrated. Such probabilistic considerations are familiar in the case of Hamiltonian systems from classical statistical mechanics, and the theory is shown to extend, with appropriate modifications, to at least some kinds of attractors. The equilibrium distribution of dissipative situations nevertheless differs from the Hamiltonian, in that it is not easy to write down. The class of attractors satisfying Smale's axiom A has a fully worked-out statistical theory, and is the subject of active mathematical research concerned with its extension to more general attractors via the notion of Liapunov characteristic exponents.