Uncertainty dynamics and predictability in chaotic systems

An initial uncertainty in the state of a chaotic system is expected to grow even under a perfect model; the dynamics of this uncertainty during the ea rly stages of its evolution are investigated. A variety of 'error growth' s tatistics are contrasted, illustrating their relative strengths when applie d to chaotic systems, all within a perfect-model scenario. A procedure is i ntroduced which can establish the existence of regions of a strange attract or within which all infinitesimal uncertainties decrease with time. It is p roven that such regions exist in the Lorenz attractor, and a number of prev ious numerical observations are interpreted in the light of this result; si milar regions of decreasing uncertainty exist in the Ikeda attractor. It is proven that no such regions exist in either the Rossler system or the Moor e-Spiegel system. Numerically strange attractors in each of these systems a re observed to sample regions of state space where the Jacobians have eigen values with negative real parts, yet when the Jacobians are not normal matr ices this does not guarantee that uncertainties will decrease. Discussions of predictability often focus on the evolution of infinitesimal uncertainti es; clearly as long as an uncertainty remains infinitesimal it cannot pose a limit to predictability. To reflect realistic boundaries, any proposed 'l imit of predictability' must be defined with respect to the nonlinear behav iour of perfect ensembles. Such limits may vary significantly with the init ial state of the system, the accuracy of the observations, and the aim of t he forecaster. Perfect-model analogues of operational weather forecasting e nsemble schemes with finite initial uncertainties are contrasted both with perfect ensembles and uncertainty statistics based upon the dynamics infini tesimal uncertainties.