Large-Scale Contributions to Spatiotemporal Chaos

Complex spatiotemporal behavior in nonlinear spatially extended systems has received extensive recent study, but is less well-understood than complex dynamics in finite-dimensional dynamical systems. We characterize some aspects of spatiotemporal chaos (STC) in a class of one-dimensional partial differential equations including the Kuramoto-Sivashinsky (KS) equation, focussing in particular on the essential contribution of large spatial scales, which act as a Gaussian ``heat bath'' maintaining the spatiotemporal disorder. We illustrate our conclusions by discussing wavelet-based numerical experiments; the construction of an effective stochastic model for the large scale dynamics; the KS equation in the presence of an additional destabilizing linear term, which displays a transition from STC to a stationary shock-like solution, due to excitation at the large scales; and a sixth-order analogue of the KS equation (the Nikolaevskii model) in which STC is maintained by the coupling to large scales.