Differential Geometry Applied to Dynamical Systems

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.