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 Λ-Cold Dark Matter cosmological models, respectively.

KCC Analysis of a One-Dimensional System During Catastrophic Shift of the Hill Function: Douglas Tensor in the Nonequilibrium Region

“…The study of the geometric invariants of a second-order ordinary differential equation is commonly called KCC theory, i.e., the general path-space theory (e.g., [Antonelli, & Bucataru, 2003;Sabȃu, 2005a;Udriste & Nicola, 2009;Neagu, 2013;Harko et al, 2016]). Because a dynamic system is often described by ordinary differential equations, KCC theory has been applied to the geometric aspects of various dynamic systems, including those for high-energy physics (e.g., [Lake & Harko, 2016;Dȃnilȃ et al, 2016]) and biological populations (e.g., [Sabȃu, 2005b;Yamasaki & Yajima, 2013]).…”

KCC Analysis of the Normal Form of Typical Bifurcations in One-Dimensional Dynamical Systems: Geometrical Invariants of Saddle-Node, Transcritical, and Pitchfork Bifurcations

“…The study of the geometric invariants of the second-order ordinary differential equation (ODE) is commonly called KCC theory (the general path-space theory of Kosambi, Cartan and Chern) (e.g., [Antonelli, & Bucataru, 2003;Sabȃu, 2005a;Yajima & Nagahama, 2007;Udriste & Nicola, 2007;Harko & Sabȃu, 2008;Balan & Neagu, 2010;Bucataru et al, 2011;Neagu, 2013]). Because the dynamic system is often described by ODEs, the KCC theory has been applied to the geometric aspects of dynamic systems in various systems, such as physical (e.g., [Boehmer & Harko, 2010;Abolghasem, 2012;Harko et al, 2015;Gupta & Yadav, 2016;Lake & Harko, 2016;Dȃnilȃ et al, 2016]), biological (e.g., [Antonelli & Bucataru, 2001;Antonelli et al, 2002;Nicola & Balan, 2004;Sabȃu, 2005b;Yamasaki & Yajima, 2013;Antonelli et al, 2014]), and general phenomena (e.g., [Udriste & Nicola, 2009;Harko et al, 2016]).…”

Dynamical behavior and Jacobi stability analysis of wound strings

“…The system of differential equations describing the deviations of the whole trajectory of a dynamical system, with respect to perturbed trajectories, are introduced geometrically in KCC theory via the second KCC invariant [13,16], and KCC theory has been extensively applied in studies of the stability of different physical, engineering, biological, and chemical systems [17]- [33]. In the present work, we apply KCC theory to the study of fundamental, or "F -string" loops, with windings in the compact internal space predicted by string theory [34,35,36,37].…”