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We study prediction-assimilation systems, which have become routine inmeteorology and oceanography and are rapidly spreading to other areas of thegeosciences and of continuum physics. The long-term, nonlinear stability ofsuch a system leads to the uniqueness of its sequentially estimated solutionsand is required for the convergence of these solutions to the system's true,chaotic evolution. The key ideas of our approach are illustrated for alinearized Lorenz system. Stability of two nonlinear prediction-assimilationsystems from dynamic meteorology is studied next via the complete spectrum oftheir Lyapunov exponents; these two systems are governed by a large set ofordinary and of partial differential equations, respectively. The degree ofdata-induced stabilization is crucial for the performance of such a system.This degree, in turn, depends on two key ingredients: (i) the observationalnetwork, either fixed or data-adaptive; and (ii) the assimilation method.

Data assimilation as a nonlinear dynamical systems problem: Stability and convergence of the prediction-assimilation system