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Journal of Dynamical Systems and Geometric Theories

(Since 2002)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0057·ISSN (Print): 1726-037X
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The Journal of Dynamical Systems and Geometric Theories (JDSGT) is a world leading journal publishing high quality, rigorously peer-reviewed original research on dynamical systems and geometry, including the interactions between these two subjects and interdisciplinary research with other branches of knowledge since 2003. Topics published by the Journal include but are not limited to: Random dynamical systems Geometry and physics Dynamical systems with hyperbolic behaviour Real and complex geometry Ergodic theory Distance geometry Topological dynamics Global differential geometry Infinite-dimensional Hamiltonian systems Symplectic geometry, contact geometry The journal considers original research articles, survey articles, and book reviews for publication. Responses to articles and correspondence will also be considered at the Chief Editor’s discretion.

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Journal of Dynamical Systems and Geometric Theories
Open Access Original Articles

Jacobi Stability of Linearized Geometric Dynamics

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pp. 161–173Vol. 7Issue 2June 2013DOI: 10.1080/1726037X.2009.10698571XML
Published Online:
03 Jun 2013
Article type:
Original Articles
Language:
EN
Article no.:
1726037X.2009.10698571
Pages:
161–173

Abstract

This paper is dedicated to the study of geometric dynamics (an Euler-Lagrange prolongation of a flow on a Riemannian manifold) from the point of view of KCC theory, Jacobi stability and Lyapounov stability. Section 1 recalls the geometrical roots of Jacobi stability and announces the subject of the paper. Section 2 introduces the variational ODEs (Jacobi fields ODEs), the KCC differential invariants for a second order ODE system, and defines the Jacobi stability. Section 3 studies the KCC differential invariants associated to geometric dynamics. Section 4 describes various linearizations of geometric dynamics. Section 5 studies the Jacobi stability for linearized geometric dynamics around a stationary point of the field. Section 6 shows that the linearized geometric dynamics around a critical point of the energy can be Jacobi stable or unstable. Section 7 proves the Lyapounov instability of Jacobi fields ODEs along geometric dynamics trajectories.

Keywords

KCC theorylinearized geometric dynamicsJacobi stabilityLyapounov stability

References

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