Advances in Theoretical and Mathematical Physics

Volume 16 (2012)

Number 4

Jacobi stability analysis of dynamical systems—applications in gravitation and cosmology

Pages: 1145 – 1196

DOI: http://dx.doi.org/10.4310/ATMP.2012.v16.n4.a2

Authors

C. G. Böhmer (Department of Mathematics and Institute of Origins, University College London)

T. Harko (Department of Mathematics, University College London)

S. V. Sabau (Department of Human Science and Informatics, Tokai University, Sapporo, Japan)

Abstract

The Kosambi–Cartan–Chern (KCC) theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach, one describes the evolution of a dynamical system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a non-linear connection and a Berwald-type connection to the dynamical system, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In the present paper, we review the basic mathematical formalism of the KCC theory, and present some specific applications of this method in general relativity, cosmology and astrophysics. In particular, we investigate the Jacobi stability of the general relativistic static fluid sphere with a linear barotropic equation of state, of the vacuum in the brane world models, of a dynamical dark energy model, and of the Lane–Emden equation, respectively. It is shown that the Jacobi stability analysis offers a powerful and simple method for constraining the physical properties of different systems, described by second-order differential equations.

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