Stretching-Based Diagnostics in a Differential Geometry Setting

TL;DR

This paper proposes a new, geometry-based approach to identify slow invariant manifolds (SIMs) in dynamical systems, extending beyond traditional slow-fast systems and ensuring coordinate invariance through differential geometry techniques.

Contribution

It introduces a covariant, differential geometry framework for defining and detecting SIMs, generalizing previous methods and addressing issues of non-uniqueness and applicability.

Findings

The geometric approach captures SIMs as invariant manifolds in phase space.

The method is coordinate-independent, ensuring robustness across different representations.

Preliminary results show promising alignment with known SIMs in example systems.

Abstract

The identification of slow invariant manifolds (SIMs) is an essential part in model-order reduction for reactive systems. The mathematical definition of the SIM by Fenichel can be considered unsatisfactory, because it is only applicable to so-called slow-fast system and does not provide the uniqueness of the SIM. Observing the phase space of the dynamical system (not necessarily a slow-fast system), the SIM becomes a geometric object which attracts trajectories, resulting in a bundling behavior. We aim to find a more general definition of the SIM, guided by the prior observations in phase space within the field of differential geometry. This setting provides one major benefit: All quantities are formulated covariantly, i.e. they are independent of the coordinate choice. A recent work by Heiter and Lebiedz \cite{heiter} translates the invariance property to vanishing sectional curvatures in the extended phase space.