  The Kosambi-Cartan-Chern (KCC) theory represents a powerful mathematical
method for the investigation of the properties of dynamical systems. The KCC
theory introduces a geometric description of the time evolution of a dynamical
system, with the solution curves of the dynamical system described by methods
inspired by the theory of geodesics in a Finsler spaces. The evolution of a
dynamical system is geometrized by introducing a non-linear connection, which
allows the construction of the KCC covariant derivative, and of the deviation
curvature tensor. In the KCC theory the properties of any dynamical system are
described in terms of five geometrical invariants, with the second one giving
the Jacobi stability of the system. Usually, the KCC theory is formulated by
reducing the dynamical evolution equations to a set of second order
differential equations. In the present paper we introduce and develop the KCC
approach for dynamical systems described by systems of arbitrary
$n$-dimensional first order differential equations. We investigate in detail
the properties of the $n$-dimensional autonomous dynamical systems, as well as
the relationship between the linear stability and the Jacobi stability. As a
main result we find that only even-dimensional dynamical systems can exhibit
both Jacobi stability and instability behaviors, while odd-dimensional
dynamical systems are always Jacobi unstable, no matter their Lyapunov
stability. As applications of the developed formalism we consider the
geometrization and the study of the Jacobi stability of the complex dynamical
networks, and of the $\Lambda $-Cold Dark Matter cosmological models,
respectively.
