A Computational Method to Extract Macroscopic Variables and Their Dynamics in Multiscale Systems

This paper introduces coordinate-independent methods for analyzing multiscale dynamical systems using numerical techniques based on the transfer operator and its adjoint. In particular, we present a method for testing whether an arbitrary dynamical system exhibits multiscale behavior and for estimating the time-scale separation. For systems with such behavior, we establish techniques for analyzing the fast dynamics in isolation, extracting slow variables for the system, and accurately simulating these slow variables at a large time step. We illustrate our method with numerical examples and show how the reduced slow dynamics faithfully represents statistical features of the full dynamics which are not coordinate dependent.