Abstract

An appealing idea of modem dynamics is that the complicated and apparently stochastic time behavior of large and even infinite dimensional nonlinear systems is in fact a manifestation of a deterministic flow on a low dimensional chaotic attractor. If the system is indeed low dimensional, it is natural to ask whether one can identify the physical characteristics such as the spatial structure of those few active modes which dominate the dynamics. The key idea is that each of these structures is a natural asymptotic state that, by virtue of the various force balances in the governing equations, developes an identity which does not easily decay or disperse away. Instead they persist and experience temporal chaos of at least two distinct types - a mild ’’phase” or more violent "amplitude” turbulence. Integrable soliton equations provide systems with many of the same qualitative properties of their finite dimensional integrable counterparts; phase space is foliated by nested tori, etc. It is fairly natural, then, to imagine that the effects of external influences (forcing, damping, coupling to other systems) will be similar to the finite dimensional case, and that the perturbed phase space will consist of a mosaic of islands of integrability and areas of stochasticity.

© 1985 Optical Society of America