Model Reduction and Fluctuation Response for Two-Timescale Systems

The purpose of this work is to study an application of the averaging method for the model reduction of chaotic two-timescale systems of ordinary differential equations. We focus on two practical implementations of the averaging method: first a simple implementation averaging with respect to a single invariant measure, then a second implementation which includes a linear response closure term derived from the fluctuation-dissipation theorem. Particular emphasis is placed on the ability of the reduced systems to capture the long-time statistics and ensemble perturbation response of the slow variables in the full system, which is not guaranteed by the averaging formalism. We present some theory in the general case regarding criteria for the similarity of the perturbation responses, then these methods are applied to a Lorenz 96 system for linear and higher-order coupling between slow and fast systems. From these numerical experiments we show that the addition of the linear response closure term greatly improves the reduced model's accuracy in capturing the dynamical and statistical behavior of the full system and could be useful in practice in the presence of computational constraints.