Slow invariant manifolds as curvature of the flow of dynamical systems

TL;DR

This paper introduces a geometric method using flow curvature to analytically determine slow invariant manifolds in high-dimensional dynamical systems, simplifying existing approaches without relying on eigenvectors or asymptotic expansions.

Contribution

It presents a novel flow curvature method that generalizes existing theories and provides a straightforward way to find slow manifolds analytically in complex systems.

Findings

Flow curvature directly yields the analytical equations of slow manifolds.

The method generalizes Tangent Linear System Approximation.

It encompasses Geometric Singular Perturbation Theory.

Abstract

Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the curvature of the trajectory curves of any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of high-dimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the so-called Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chua's piecewise linear and cubic models of dimensions three, four and five will be provided as tutorial examples exemplifying this method as well as those of high-dimensional dynamical systems.