A variational principle for computing slow invariant manifolds in dissipative dynamical systems

TL;DR

This paper introduces a variational method for accurately computing slow invariant manifolds in dissipative systems, combining theoretical proofs and numerical experiments to validate its effectiveness across models.

Contribution

It develops a variational principle-based approach for identifying slow invariant manifolds, with analytical and numerical validation for linear, nonlinear, and chemical reaction models.

Findings

The approach asymptotically finds exact manifolds in the limit of infinite time horizon or spectral gap.

Numerical results confirm the method's effectiveness on various models.

The method applies to higher-dimensional chemical reaction mechanisms.

Abstract

A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing such manifolds for kinetic models using trajectory optimization. The corresponding objective functional reflects a variational principle that characterizes trajectories on, respectively near, slow invariant manifolds. For a two-dimensional linear system and a common nonlinear test problem we show analytically that the variational approach asymptotically identifies the exact slow invariant manifold in the limit of both an infinite time horizon of the variational problem with fixed spectral gap and infinite spectral gap with a fixed finite time horizon. Numerical results for the linear and nonlinear model problems as well as a more realistic higher-dimensional chemical reaction mechanism are presented.