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Differential Geometry Applied to Dynamical Systems
Abstract
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.