In this paper we deal with the cosmological dynamics of Randall-Sundrum bra
ne-world type scenarios in which the five-dimensional Weyl tensor has a non
vanishing projection onto the three-brane, where matter fields are confined
. Using dynamical systems techniques, we study how the state space of Fried
mann-Lemaitre-Robertson-Walker and Bianchi type I cosmological models is af
fected by the bulk Weyl tensor, focusing on the differences that appear wit
h respect to general relativity and also Randall-Sundrum cosmological scena
rios without the Weyl tensor contribution.