A trajectory-free framework for analysing multiscale systems

TL;DR

This paper introduces a computationally efficient, trajectory-free framework using transfer and Koopman operators to analyze multiscale systems, identify time-scale separation, and extract reduced slow dynamics.

Contribution

It presents novel algorithms that do not rely on trajectory integration for detecting multiscale behavior and estimating time-scale separation in dynamical systems.

Findings

Algorithms successfully identify multiscale dynamics.

Efficient estimation of time-scale separation.

Effective extraction of reduced slow dynamics.

Abstract

We develop algorithms built around properties of the transfer operator and Koopman operator which 1) test for possible multiscale dynamics in a given dynamical system, 2) estimate the magnitude of the time-scale separation, and finally 3) distill the reduced slow dynamics on a suitably designed subspace. By avoiding trajectory integration, the developed techniques are highly computationally efficient. We corroborate our findings with numerical simulations of a test problem.