Spatially localized models of extended systems

We investigate the construction of low-dimensional spatially localized models of extended systems. Specifically, the Kuramoto-Sivashinsky (KS) equation on large one-dimensional domains displays spatiotemporally complex dynamics that are remarkably well-localized in both real and Fourier space, as demonstrated by a (spline) wavelet representation. We show how wavelet projections may be used to construct various localized, relatively low-dimensional models of KS spatiotemporal chaos. There is persuasive evidence that short, periodized systems, internally forced at their largest scales, form minimal models for chaotic dynamics in arbitrarily large domains. Such models assist in the understanding of extended systems.