A computational method to extract macroscopic variables and their dynamics in multiscale systems

TL;DR

This paper presents coordinate-independent numerical methods for analyzing multiscale dynamical systems, enabling the detection of multiscale behavior, extraction of slow variables, and efficient simulation of slow dynamics.

Contribution

It introduces a novel approach using transfer operators to identify multiscale behavior and extract slow variables in a coordinate-independent manner.

Findings

Successfully detects multiscale behavior in systems

Accurately extracts slow variables for reduced modeling

Efficiently simulates slow dynamics with preserved statistical features

Abstract

This paper introduces coordinate-independent methods for analysing 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 behaviour and for estimating the time-scale separation. For systems with such behaviour, we establish techniques for analysing 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.