  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.
