Statistical topography and Lyapunov exponents in stochastic dynamical systems

This article discusses the relationship between the statistical description of stochastic dynamical systems based on the ideas of statistical topography and the traditional analysis of Lyapunov stability of dynamical systems with the use of the Lyapunov characteristic indices (Lyapunov exponents). As an illustration, some coherent phenomena are considered that occur with a probability of unity, i.e., in almost all realizations of the stochastic systems. Among such phenomena are the diffusion and clustering of a passive tracer in random hydrodynamic flows, the dynamic localization of plane waves in layered random media, and the emergence of caustic patterns of the wave field in multidimensional random media.