Abstract

In the first part while 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.

In the second part it will be established that the flow curvature method provides linear invariant manifolds of n-dimensional dynamical systems, i.e., first extactic algebraic manfiolds and so constitutes a particular case of Lagutinskii′s method.

Then, many applications are proposed to exemplify the efficiency of this method for high-dimensional dynamical systems.