points does not represent a stable isotropic state except if $\ell\rightarrow 1$. Then it tends toward the previous one. However, we will see below that the value $\ell=1$ does not allow for isotropisation. Thus we discard it.\
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*Monotonic functions*\
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We rewrite the equation (\[hamp\]) with the normalised variables: $$\label{hamp1}
\dot{H}=-H(72y^2+24w^2)$$ Hence the Hamiltonian is a monotonic function of $\Omega$ with a constant sign. Then, from the lapse function expression (\[lapse\]), we deduce that $\Omega$ is a monotonic function of the proper time $t$. Therefore, if the Hamiltonian is initially positive (negative), $\Omega\rightarrow -\infty$ corresponds to late time(early time). We will not consider the case $\Omega\rightarrow +\infty$ since we will show below that it does not lead to isotropy. We conclude that late times isotropisation initially needs $H>0$: this is the only necessary initial condition for this behaviour. Moreover, the Hamiltonian being of constant sign, it is the same for the variables $y$ and $w$.\
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*Asymptotic behaviours*\
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We wish to evaluate the behaviours of some quantities in the neighbourhood of the equilibrium points $(x_{+},x_{-},y,z,w)=(0,0,\pm \sqrt{3-\ell^2}(6\sqrt{2})^{-1},\ell/6,0)$, i.e. when we approach isotropy in $\Omega\rightarrow -\infty$. Approximating (\[eq5\]) near equilibrium by $w(1-\ell^2)$, we find that asymptotically $w$ behaves as $e^{(1-\ell^2)\Omega}$.

