## Jason Frank (Utrecht University)

### Notes on Synchronization, Shadowing and Data Assimilation

The research of Pecora and Carroll (1990) drew attention to the concept of synchronization of chaotic systems: some projection of the solution of one system is used to drive another, such that the state of the first system can be completely observed. Much of the original research in this area addressed message encryption; however, it is attractive as an approach to data assimilation, where one attempts to deduce a trajectory of a system by combining a dynamical model with instrumental observations. (Synchronization also seems reminiscent of the embedding theorem by Takens.) It has long been established that numerical integrations of hyperbolic systems, while rapidly diverging from the trajectory through the initial condition, nevertheless can be shown to shadow some trajectory of the original system for very long time intervals. The proof of this using the Newton-Kantorovich theorem suggests a different approach to data assimilation. In this talk I will review some of these concepts and discuss numerical results from a new data assimilation algorithm.