Phase space analysis of quintessence fields trapped in a Randall-Sundrum braneworld: A refined study

In this paper, we investigate, from the perspective of dynamical systems, the evolution of a scalar field with an arbitrary potential trapped in a Randall-Sundrum's braneworld of type II. We consider an homogeneous and isotropic Friedmann-Robertson-Walker brane filled also with a perfect fluid. Center manifold theory is employed to obtain sufficient conditions for the asymptotic stability of the de Sitter solution. We obtain conditions on the potential for the stability of scaling solutions as well for the stability of the scalar-field-dominated solution. We prove the fact that there are not late-time attractors with 5D-modifications (they are saddle like). This fact correlates with a transient primordial inflation. In the particular case of a scalar field with the potential V = V ₀e ^χ + Λ, we prove that for χ < 0 the de Sitter solution is asymptotically stable. However, for χ > 0, the de Sitter solution is unstable (of saddle type).