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

TL;DR

This paper extends the Kosambi-Cartan-Chern (KCC) theory to analyze higher-dimensional first-order dynamical systems, revealing that only even-dimensional systems can exhibit both Jacobi stability and instability, with applications in complex networks and cosmology.

Contribution

The paper develops a generalized KCC framework for arbitrary n-dimensional first-order systems, linking linear and Jacobi stability, and explores its implications for complex networks and cosmological models.

Findings

Only even-dimensional systems can show both Jacobi stability and instability.

Odd-dimensional systems are always Jacobi unstable regardless of Lyapunov stability.

Application to complex networks and cosmological models demonstrates the framework's versatility.

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.