Identification and Computation of Slow Manifolds Using the Isostable Coordinate System

2025/07/18 by Dan Wilson, Wilson, Dan
Engineering · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mechanical and Optical Resonators #Model Reduction and Neural Networks #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2507.13997

openalex publication_date 2025/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Existence and uniqueness of global Koopman eigenfunctions for stable\n fixed points and periodic orbits
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Cited by

• Reduced Order Modeling of Nonlinear Dynamical Systems Using Slow Manifolds

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