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A unified framework for studying parametric and initial condition uncertainty in dynamical systems is developed. The notion of input measure of an observable is defined and its propagation to output measure of the observable is studied by means of transfer operators. Uncertainty of these measures is defined in terms of their cumulative probability distributions. The developed formalism is illustrated through an analysis of the effect of pitchfork bifurcation on uncertainty. General results on uncertainty for dynamical systems on an infinite time horizon are derived.
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Mezić et al. (2004) studied this question.