Jacobi stability analysis of the Lorenz system

We perform the study of the stability of the Lorenz system by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. The Lorenz model plays an important role for understanding hydrodynamic instabilities and the nature of the turbulence, also representing a nontrivial testing object for studying nonlinear effects. The KCC theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach, we describe the evolution of the Lorenz system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a nonlinear connection and a Berwald type connection, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in) stability is a natural generalization of the (in) stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In order to apply the KCC theory, we reformulate the Lorenz system as a set of two second-order nonlinear differential equations. The geometric invariants associated to this system (nonlinear and Berwald connections), and the deviation curvature tensor, as well as its eigenvalues, are explicitly obtained. The Jacobi stability of the equilibrium points of the Lorenz system is studied, and the condition of the stability of the equilibrium points is obtained. Finally, we consider the time evolution of the components of the deviation vector near the equilibrium points.