DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS - Archive ouverte HAL Access content directly
Books Year : 2009

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

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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.
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Dates and versions

hal-01101601 , version 1 (09-01-2015)

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  • HAL Id : hal-01101601 , version 1

Cite

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⟩
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