Towards differential geometric characterization of slow invariant manifolds in extended state space: Sectional Curvature and Flow Invariance

TL;DR

This paper introduces a differential geometric framework, including sectional curvature, to characterize slow invariant manifolds in extended phase space, aiming for a coordinate-independent and intrinsic understanding of their properties.

Contribution

It proposes a geometric approach to identify and analyze slow invariant manifolds using curvature and invariance conditions, advancing model reduction techniques.

Findings

Derived a necessary geometric condition for SIAM identification

Applied the framework to example systems demonstrating its effectiveness

Explored potential for sufficient conditions based on variational principles

Abstract

Some model reduction techniques for multiple time-scale dynamical systems make use of the identification of low dimensional slow invariant attracting manifolds (SIAM) in order to reduce the dimensionality of the phase space by restriction to the slow flow. The focus of this work is on a proposition and discussion of a general viewpoint using differential geometric concepts for submanifolds to deal with slow invariant manifolds in an extended phase space. The motivation is a coordinate independent formulation of the manifold properties and its characterization problem treating the manifold as intrinsic geometric object. We formulate a computationally verifiable necessary condition for the slow invariant manifold graph stated in terms of a differential geometric view on the invariance property. Its application to example systems is illustrated. In addition, we present some ideas and investigations concerning the search for sufficient, differential geometric conditions characterizing slow invariant manifolds based on our previously developed variational principle.