Asymptotic behavior of a scalar field with an arbitrary potential
  trapped on a Randall-Sundrum's braneworld: the effect of a negative dark
  radiation term on a Bianchi I brane
  In this work we present a phase space analysis of a quintessence field and a
perfect fluid trapped in a Randall-Sundrum's Braneworld of type 2. We consider
a homogeneous but anisotropic Bianchi I brane geometry. Moreover, we consider
the effect of the projection of the five dimensional Weyl tensor onto the
three-brane in the form of a negative Dark Radiation term. For the treatment of
the potential we use the "Method of $f$-devisers" that allows investigating
arbitrary potentials in a phase space. We present general conditions on the
potential in order to obtain the stability of standard 4D and non-standard 5D
de Sitter solutions, and we provide the stability conditions for both scalar
field-matter scaling solutions, scalar field-dark radiation solutions and
scalar field-dominated solutions. We find that the shear-dominated solutions
are unstable (particularly, contracting shear-dominated solutions are of saddle
type). As a main difference with our previous work, the traditionally
ever-expanding models could potentially re-collapse due to the negativity of
the dark radiation. Additionally, our system admits a large class of static
solutions that are of saddle type. These kinds of solutions are important at
intermediate stages in the evolution of the universe, since they allow the
transition from contracting to expanding models and viceversa. New features of
our scenario are the existence of a bounce and a turnaround, which lead to
cyclic behavior, that are not allowed in Bianchi I branes with positive dark
radiation term. Finally, as specific examples we consider the potentials
$V\propto\sinh^{-\alpha}(\beta\phi)$ and $V\propto\left[\cosh\left(\xi\phi
\right)-1\right]$ which have simple $f$-devisers.
