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 alocal level in such systems, the macroscopic state of the system typically remains periodic or stationary. This implies that the dimensionD of chaotic (strange) attractors must diverge with the linear sizeL of the system likeD(L/C)^d ind space dimensions, where (<) 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.