Differential Geometry Applied to Dynamical Systems

4 February 2010
Prof. Jean-Marc Ginoux
This work 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 co-dimension one, centre 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. Moreover, the concept of curvature of trajectory curves applied to classical dynamical systems such as Lorenz and Rossler models enabled to highlight one-dimensional invariant sets, i.e. curves connecting fixed points which are zero-dimensional invariant sets. Such "connecting curves" provide information about the structure of the attractors and may be interpreted as the skeleton of these attractors. Many examples are given in dimension three and more.
  • Applied Dynamical Systems and Inverse Problems Seminar