Abstract: We study the synchronization of two spatially extended dynamical systems where the models have imperfections. We show that the synchronization error across space can be visualized as a rough surface governed by the Kardar-Parisi-Zhang equation with both upper and lower bounding walls corresponding to nonlinearities and model discrepancies, respectively. Two types of model imperfections are considered: parameter mismatch and unresolved fast scales, finding in both cases the same qualitative results. The consistency between different setups and systems indicates that the results are generic for a wide family of spatially extended systems. Identical chaotic systems are able to perfectly synchronize when coupled. This phenomenon is very attractive from a theoretical perspective and has a tremendous potential for technological applications, like for instance, secure optical communications. Chaotic synchronization of spatially extended dynamical systems plays also a fundamental role in forecasting applications and observation data assimilation in geoscience. Unfortunately, dynamical systems are far from identical in practical applications and synchronization of high dimensional systems in the real world becomes severely hampered by imperfections like parameter mismatches and unresolved scales. Understanding the bounds to synchronization for imperfect systems, the effect of finite non-small parameter deviations, and limited resolution on the statistics of the synchronization error is essential for real applications. The theoretical challenge is to describe synchronization of imperfect models under the umbrella of a generic stochastic theory that describes the most outstanding features in a model–independent fashion.

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