Stochastic models of chaotic systems

Nonlinear dynamical systems, although strictly deterministic, often exhibit chaotic behavior which appears to be random. The determination of the probabilistic properties of such systems is, in general, an open problem. Closure approximations for moment expansion methods have been unsatisfactory. More successful has been approximation on the dynamics level by the use of linear stochastic models that attempt to generate the probabilistic properties of the original nonlinear chaotic system as closely as possible. Examples are reviewed of this approach to simple nonlinear systems, to turbulence, and to large-eddy simulation. A stochastic model that simulates the transient energy spectrum of the global atmosphere is developed