  In this paper we investigate, from the dynamical systems perspective, the
evolution of an scalar field with arbitrary potential trapped in a
Randall-Sundrum's Braneworld of type 2. We consider an homogeneous but
anisotropic Bianchi I (BI) brane filled also with a perfect fluid. We also
consider the effect of the projection of the five-dimensional 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
de Sitter solution with standard 4D behavior. We also prove that there are not
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.

General Relativity and Quantum Cosmology;High Energy Physics - Theory