On Differential Geometric Formulations of Slow Invariant Manifold Computation: Geodesic Stretching and Flow Curvature

TL;DR

This paper introduces a coordinate-free, differential geometric approach to approximate slow invariant manifolds in dynamical systems, utilizing Riemannian geometry and geodesic flows to improve model reduction techniques.

Contribution

It develops an intrinsic, tensor-based criterion for approximating normally attracting invariant manifolds, extending existing diagnostics to a covariant geometric framework.

Findings

The method successfully approximates slow invariant manifolds in test models.

It generalizes stretching diagnostics and flow curvature methods using Riemannian geometry.

The approach is coordinate-free and applicable without explicit time-scale separation.

Abstract

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.