Slow Invariant Manifolds of Slow-Fast Dynamical Systems

Slow-fast dynamical systems, i.e. singularly or nonsingularly 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.

Automated summary

This paper classifies methods for approximating slow-fast dynamical systems equations into two categories: singular perturbation-based methods and curvature-based methods. It also proves the equivalence between methods within the same category and between the two categories, and analyzes and compares each method to demonstrate the efficiency of the Flow Curvature Method.