In this paper, the authors study the problem of finding the best approximate aggregation of dynamical systems, by considering the dynamics for macrovariables such that a certain criterion of inconsistency between the aggregated and original systems is minimized. First, the aggregation giving the least square deviation of the vector fields is obtained for any nonlinear dynamical system. Second, the best aggregation of linear systems around equilibria is examined by minimization of various criteria, such as (1) the difference in vector fields, (2) the difference in variables at a certain time point, (3) the difference in temporally averaged variables, and (4) the temporal average of square difference in variables. Finally, the determination of parameters in nonlinear dynamical systems by sequential application of several optimality criteria is discussed. In short, the best aggregated system greatly depends on the choice of criterion, especially with regard to the selection of the time horizon and of the weighting for the initial state.
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