Bridging the Gap between Koopmanism and Response Theory: Using Natural Variability to Predict Forced Response

Abstract: The fluctuation-dissipation theorem is a cornerstone result in statistical mechanics that can be used to translate the statistics of the free natural variability of a system into information on its forced response to perturbations. By combining this viewpoint on response theory with the key ingredients of Koopmanism, it is possible to deconstruct virtually any response operator into a sum of terms, each associated with a specific mode of natural variability of the system. This dramatically improves the interpretability of the resulting response formulas. We show here on three simple yet mathematically meaningful examples how to use the Extended Dynamical Mode Decomposition (EDMD) algorithm on an individual trajectory of the system to compute with high accuracy correlation functions as well as Green functions associated with acting forcings. This demonstrates the great potential of using Koopman analysis for the key problem of evaluating and testing the sensitivity of a complex system.