Phase space analysis of quintessence fields trapped in a Randall-Sundrum braneworld: Anisotropic Bianchi i brane with a positive dark radiation term

In this paper we investigate, from the dynamical systems perspective, the evolution of a scalar field with an arbitrary potential trapped in a Randall-Sundrum's braneworld of type 2. We consider a homogeneous but anisotropic Bianchi I (BI) brane filled with a perfect fluid. We also consider the effect of the projection of the five-dimensional (5D) Weyl tensor onto the three-brane in the form of a positive dark radiation term. Using the center manifold theory, we obtain sufficient conditions for the asymptotic stability of the de Sitter solution with standard 4D behavior. We also prove that there are no late-time de Sitter attractors with 5D modifications since they are always saddle-like. This fact correlates with a transient primordial inflation. We present here sufficient conditions on the potential for the stability of the scalar field-matter scaling solution, the scalar-field-dominated solution and the scalar field-dark radiation scaling solution. We illustrate our analytical findings using a simple f-deviser as a toy model. All these results are generalizations of our previous results obtained for FRW branes.