Approximating normally attracting invariant manifolds using trajectory-based optimization

TL;DR

This paper analyzes a trajectory-based optimization method for approximating normally attracting invariant manifolds in complex dynamical systems, providing a rigorous foundation and extending its applicability to higher-dimensional manifolds with nonlinear constraints.

Contribution

It offers a rigorous explanation of the method's effectiveness and extends its use to higher-dimensional, nonuniformly normally attracting invariant manifolds with nonlinear constraints.

Findings

Provides a coordinate-free formulation on Riemannian manifolds.

Shows the method approximates nonuniformly normally attracting orbits.

Extends the approach to higher-dimensional invariant manifolds.

Abstract

The numerical simulation of realistic reactive flows is a major challenge due to the stiffness and high dimension of the corresponding kinetic differential equations. Manifold-based model reduction techniques address this problem by projecting the full phase space onto manifolds of slow motion, which capture the system's long-term behavior. In this article we study the trajectory-based optimization approach by Lebiedz (2004), which determines these manifolds as minimizers of an appropriate entropy functional. Similar to other methods in this field, this approach is based on physical and geometric intuition and was tested on several models. This article provides a rigorous explanation for its effectiveness, showing how it approximates nonuniformly normally attracting orbits. It also outlines how the method can be utilized to approximate nonuniformly normally attracting invariant manifolds of higher dimension. Throughout the article we use a coordinate-free formulation on a Riemannian manifold. This is especially useful for systems subject to nonlinear constraints, e.g., adiabatic constraints.