A Monte Carlo approach to understanding the impacts of initial-condition uncertainty, model uncertainty, and simulation variability on the predictability of chaotic systems: Perspectives from the one-dimensional logistic map

Abstract

The predictability of the logistic map is investigated for the joint impact of initial-condition (IC) and model uncertainty (bias + random variability) as well as simulation variability. To this end, Monte Carlo simulations are carried out where IC bias is varied in a wide range of 10−15–10−3, and, similarly, model bias is introduced in comparable range. It is found that while the predictability limit of the logistic map can be continuously extended by reducing IC bias, the introduction of the model bias imposes an upper limit to the predictability limit beyond which further reductions in IC bias do not lead to an extension in the predictability limit, effectively restricting the feasible joint space spanned by the IC-model biases. It is further observed that imposing a lower limit to the allowed variability among ensemble solutions (so as to prevent the ensemble variability from collapse) results in a similar constraint in the joint IC-model-bias space; but this correspondence breaks down when the imposed variability limit is too high (∼x>0.7 for the logistic map). Finally, although increasing the IC random variability in an ensemble is found to consistently extend the allowed predictability limit of the logistic map, the same is not observed for model parameter random variability. In contrast, while low levels of model parameter variability have no impact on the allowed predictability limit, there appears to be a threshold at which an abrupt transition occurs toward a distinctly lower predictability limit.