Slow Invariant Manifolds of Slow-Fast Dynamical Systems

TL;DR

This paper classifies and compares methods for approximating slow invariant manifolds in slow-fast dynamical systems, demonstrating the equivalence of different approaches and highlighting the efficiency of the Flow Curvature Method through examples.

Contribution

It provides a classification of existing methods into two categories, proves their equivalence within and across categories, and analyzes the Flow Curvature Method's effectiveness.

Findings

Flow Curvature Method is efficient for approximating slow invariant manifolds.

Singular perturbation-based and curvature-based methods are equivalent within their categories.

The paper demonstrates these concepts with Van der Pol and Lorenz systems.

Abstract

Slow-fast dynamical systems, i.e., singularly or non-singularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating their equations. This paper aims, on the one hand, to propose a classification of the most important of them into two great categories: singular perturbation-based methods and curvature-based methods, and on the other hand, to prove the equivalence between any methods belonging to the same category and between the two categories. Then, a deep analysis and comparison between each of these methods enable to state the efficiency of the Flow Curvature Method which is exemplified with paradigmatic Van der Pol singularly perturbed dynamical system and Lorenz slow-fast dynamical system.