Jacobi stability of the vacuum in the static spherically symmetric brane world models

We analyze the stability of the structure equations of the vacuum in the brane world models, by using both the linear (Lyapunov) stability analysis, and the Jacobi stability analysis, the Kosambi-Cartan-Chern theory. In the brane world models the four-dimensional effective Einstein equations acquire extra terms, called dark radiation and dark pressure, respectively, which arise from the embedding of the three-brane in the bulk. Generally, the spherically symmetric vacuum solutions of the brane gravitational field equations have properties quite distinct as compared to the standard black hole solutions of general relativity. We close the structure equations by assuming a simple linear equation of state for the dark pressure. In this case the vacuum is Jacobi stable only for a small range of values of the proportionality constant relating the dark pressure and the dark radiation. The unstable trajectories on the brane behave chaotically, in the sense that after a finite radial distance it would be impossible to distinguish the trajectories that were very near each other at an initial point. Hence the Jacobi stability analysis offers a powerful method for constraining the physical properties of the vacuum on the brane.