Jacobi stability analysis of dynamical systems - applications in gravitation and cosmology. (English) Zbl 1273.83177

Böhmer, C. G.; Harko, T.; Sabau, S. V.

Adv. Theor. Math. Phys. 16, No. 4, 1145-1196 (2012).

Summary: The Kosambi-Cartan-Chern (KCC) theory [D. D. Kosambi, Math. Z. 37, 608–618 (1933; Zbl 0007.23004)] 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.

MSC:

83F05	Relativistic cosmology
37N20	Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
83C10	Equations of motion in general relativity and gravitational theory
53B40	Local differential geometry of Finsler spaces and generalizations (areal metrics)
83D05	Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
85A40	Astrophysical cosmology

Keywords:

Jacobi stability; dynamical system; cosmology; Kosambi-Cartan-Chern; KCC; geodesic; Finsler space; Lane-Emden equation

Citations:

Zbl 0007.23004