Dynamical systems with time scale separation: averaging, stochastic modelling, and central limit theorems

Abstract

Considering dynamical systems involving processes on both slow and fast time scales, we deal with two methods to obtain reduced evolution equations for the slow variables alone: While Averaging yields effective models for prediction, issues like variability might profit from stochastic modelling. Rigorous results are available only in the limit of an infinite ratio between the two time scales. In a numerical case study, we show that reduced models obtained by Averaging may possess good predictive skill even far from the region of applicability of the Averaging Theorem (time scale ratio only around 10). Stochastic modelling of the same numerical example does not predict better, but it additionally provides information on the prediction error and on the long-term variability. For a practical implementation of a stochastic model, approximation of the fast variables by Gaussian white noise is desirable. We review some recent rigorous results on Central Limit Theorems and the way how deterministic chaotic dynamical systems can approach a (stochastic) Langevin process in the limit of infinitely separated time scales. Again, the numerical example indicates their relevance also under less idealized conditions.