DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS

Abstract : This book aims to present a new approach called Flow Curvature Method that applies Differential Geometry to Dynamical Systems. Hence, for a trajectory curve, an integral of any n-dimensional dynamical system as a curve in Euclidean n-space, the curvature of the trajectory — or the flow — may be analytically computed. Then, the location of the points where the curvature of the flow vanishes defines a manifold called flow curvature manifold. Such a manifold being defined from the time derivatives of the velocity vector field, contains information about the dynamics of the system, hence identifying the main features of the system such as fixed points and their stability, local bifurcations of codimension one, center manifold equation, normal forms, linear invariant manifolds (straight lines, planes, hyperplanes). In the case of singularly perturbed systems or slow-fast dynamical systems, the flow curvature manifold directly provides the slow invariant manifold analytical equation associated with such systems. Also, starting from the flow curvature manifold, it will be demonstrated how to find again the corresponding dynamical system, thus solving the inverse problem.
Document type :
Books
Complete list of metadatas

https://hal-univ-tln.archives-ouvertes.fr/hal-01101601
Contributor : Jean-Marc Ginoux <>
Submitted on : Friday, January 9, 2015 - 10:17:15 AM
Last modification on : Friday, January 25, 2019 - 3:52:01 PM

Identifiers

  • HAL Id : hal-01101601, version 1

Collections

Citation

Jean-Marc Ginoux. DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS. ⟨Wolrd Scientific⟩, 66, 2009, Series on Nonlinear Science, L.O. Chua, 978-981-4277-14-3. ⟨hal-01101601⟩

Share

Metrics

Record views

90