Stability of nonstationary states of classical, many-body dynamical systems

We summarize recent arguments which show that for a broad class of classical, many-body dynamical model systems with short-range interactions (such as coupled maps, cellular automata, or partial differential equations), collectively chaotic states-nonstationary states wherein some Fourier amplitude varies chaotically in time-cannot occur generically. While chaos occurs ubiquitously on a local level in such systems, the macroscopic state of the system typically remains periodic or stationary. This implies that the dimension D of chaotic ("strange") attractors must diverge with the linear size L of the system like D∼(L/ξC)d in d space dimensions, where ξ (<∞) is the spatial coherence length. We also summarize recent work which demonstrates that in spatially isotropic systems that have short-range interactions and evolve (like coupled maps) in discrete time, periodic states are never stable under generic conditions. In spatially anisotropic systems, however, short-range interactions that exploit the anisotropy and so allow for the stabilization of periodic states do exist. © 1988 Plenum Publishing Corporation.