Identification and Computation of Slow Manifolds Using the Isostable Coordinate System

TL;DR

This paper develops new computational methods leveraging the isostable coordinate system and Koopman analysis to identify slow manifolds in nonlinear dynamical systems, aiding model reduction and understanding of slow dynamics.

Contribution

It introduces two novel strategies for approximating backward-time solutions on slow manifolds, addressing numerical challenges in separating fast and slow timescales.

Findings

Effective identification of slow manifolds demonstrated on various examples.

Methods extend beyond linear regimes, capturing nonlinear slow dynamics.

Potential applications in model order reduction and system analysis.

Abstract

Koopman analysis can be used to understand the dynamics of a nonlinear dynamical system in terms a linear, but generally infinite dimensional operator. The isostable coordinate system focuses on the slowest decaying principal Koopman eigenmodes. This work leverages the isostable coordinate framework in the identification of slow manifolds for dynamical systems with fixed point attractors, defined as surfaces for which the fastest decaying isostable coordinates are zero. Numerical challenges associated with separation between fast and slow timescales necessitate the development of new computational approaches to identify these slow manifolds. Two such strategies are developed which approximate backward-time solutions on the slow manifold starting near the fixed point and extending far beyond the linear regime. Application to a variety of examples illustrates the utility of these methods and their potential use for model order reduction purposes.