  We perform the study of the stability of the cosmological scalar field
models, by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern
(KCC) theory. In the KCC approach we describe the time evolution of the scalar
field cosmologies in geometric terms, by performing a "second geometrization",
by considering them as paths of a semispray. By introducing a non-linear
connection and a Berwald type connection associated to the Friedmann and
Klein-Gordon equations, five geometrical invariants can be constructed, with
the second invariant giving the Jacobi stability of the cosmological model. We
obtain all the relevant geometric quantities, and we formulate the condition of
the Jacobi stability for scalar field cosmologies in the second order
formalism. As an application of the developed methods we consider the Jacobi
stability properties of the scalar fields with exponential and Higgs type
potential. We find that the Universe dominated by a scalar field exponential
potential is in Jacobi unstable state, while the cosmological evolution in the
presence of Higgs fields has alternating stable and unstable phases. By using
the standard first order formulation of the cosmological models as dynamical
systems we have investigated the stability of the phantom quintessence and
tachyonic scalar fields, by lifting the first order system to the tangent
bundle. It turns out that in the presence of a power law potential both these
models are Jacobi unstable during the entire cosmological evolution.
