Slow Invariant Manifolds of Slow-Fast Dynamical 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.

arXiv commentarXiv admin note: text overlap with arXiv:1808.08058

References in corpus (7)

• Low-Dimensional Modelling of Dynamics via Computer Algebra
• Differential Geometry and Mechanics Applications to Chaotic Dynamical Systems
• Analysis of the accuracy and convergence of equation-free projection to a slow manifold
• Slow invariant manifolds as curvature of the flow of dynamical systems
• Connecting curves for dynamical systems
• The Slow Invariant Manifold of the Lorenz--Krishnamurthy Model
• Slow Manifold of a Neuronal Bursting Model

Cited by in corpus (1)

• Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning