General disagreement between the Geometrical Description of Dynamical In-stability -using non affine parameterizations- and traditional Tangent Dynamics

2008/07/08 by Eduardo Cuervo‐Reyes, Eduardo Cuervo-Reyes, Cuervo-Reyes, Eduardo
Mathematics · Physics and Astronomy · #37D99 #37J99 #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #math-ph #math.MP #msc:37D99 #msc:37J99

paper · pdf · doi:10.48550/arxiv.0807.1156

8 pages in preprint format or 4 pages in pre format. 0 figures

arxiv created 2008/07/08 · openalex publication_date 2008/07/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, the general disagreement of the geometrical lyapunov exponent with lyapunov exponent from tangent dynamics is addressed. It is shown in a quite general way that the vector field of geodesic spread ξ^k_G is not equivalent to the tangent dynamics vector ξ^k_T if the parameterization is not affine and that results regarding dynamical stability obtained in the geometrical framework can differ qualitatively from those in the tangent dynamics. It is also proved in a general way that in the case of Jacobi metric -frequently used non affine parameterization-, ξ^k_G satisfies differential equations which differ from the equations of the tangent dynamics in terms that produce parametric resonance, therefore, positive exponents for systems in stable regimes.

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