Chaotic Synchronization of Partial Differential Equations by Partial Driving: Framework and Selected Models

We study chaotic synchronization between two copies of a spatially extended system governed by a partial differential equation (PDE) when the second copy, the slave, receives only partial information from the first, the master: measurements at a few points, or only the low-frequency modes. After setting up the general framework, we explain why dissipation makes the amount of information required finite, and we apply the idea to three models: the Kuramoto–Sivashinsky equation, the two-dimensional Navier–Stokes equations and the complex Ginzburg–Landau equation. Two numerical experiments, one on Kuramoto–Sivashinsky with point observations and one on a two-dimensional turbulent flow observed through its large scales, show how the error decays and how the density of observations decides whether synchronization occurs. We close with applications, mainly data assimilation for weather and ocean forecasting, and with the limitations of the approach.