Kosambi-Cartan-Chern (KCC) theory for higher order dynamical systems

Abstract

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.

16 pages, no figures; new section on cosmology added; references added; accepted for publication in Int. J. Geom. Methods Mod. Phys

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• Jacobi stability of the vacuum in the static spherically symmetric brane world models
• Introduction to the application of the dynamical systems theory in the study of the dynamics of cosmological models of dark energy
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• Stability of circular geodesics in equatorial plane of Kerr spacetime
• Relevant scales for the $C$-metric with positive cosmological constant
• Jacobi Stability Analysis for Systems of ODEs Using Symbolic Computation
• Stability analysis of geodesics in dynamical Chern-Simons black holes: a geometrical perspective