Asymptotic states in brane cosmology with a nonlocal anisotropic stress

We investigate the dynamics of a Bianchi I brane Universe in the presence of a nonlocal anisotropic stress ${\mathcal{P}}_{\mu \nu }$ proportional to a “dark energy” $\mathcal{U}$. Using this ansatz for the case $\mathcal{U}>0$ we prove that if a matter on a brane satisfies the equation of state $p=(\gamma -1)\rho$ with $\gamma \le 4\mathrm{/}3$ then all such models isotropize. For $\gamma >4\mathrm{/}3$ anisotropic future asymptotic states are found. We also describe the past asymptotic regimes for this model.