  We perform the study of the stability of the Lorenz system by using the
Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. The Lorenz
model plays an important role for understanding hydrodynamic instabilities and
the nature of the turbulence, also representing a non-trivial testing object
for studying non-linear effects. The KCC theory represents a powerful
mathematical method for the analysis of dynamical systems. In this approach we
describe the evolution of the Lorenz 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, 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 order to apply the KCC theory we
reformulate the Lorenz system as a set of two second order non-linear
differential equations. The geometric invariants associated to this system
(nonlinear and Berwald connections), and the deviation curvature tensor, as
well as its eigenvalues, are explicitly obtained. The Jacobi stability of the
equilibrium points of the Lorenz system is studied, and the condition of the
stability of the equilibrium points is obtained. Finally, we consider the time
evolution of the components of the deviation vector near the equilibrium
points.
