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We analyze the stability of the structure equations of the vacuum in thebrane world models, by using both the linear (Lyapunov) stability analysis, andthe Jacobi stability analysis, the Kosambi-Cartan-Chern (KCC) theory. In thebrane world models the four dimensional effective Einstein equations acquireextra terms, called dark radiation and dark pressure, respectively, which arisefrom the embedding of the 3-brane in the bulk. Generally, the sphericallysymmetric vacuum solutions of the brane gravitational field equations, haveproperties quite distinct as compared to the standard black hole solutions ofgeneral relativity. We close the structure equations by assuming a simplelinear equation of state for the dark pressure. In this case the vacuum isJacobi stable only for a small range of values of the proportionality constantrelating the dark pressure and the dark radiation. The unstable trajectories onthe brane behave chaotically, in the sense that after a finite radial distanceit would be impossible to distinguish the trajectories that were very near eachother at an initial point. Hence the Jacobi stability analysis offers apowerful method for constraining the physical properties of the vacuum on thebrane.

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