as $\Omega\rightarrow -\infty$, the concerned variables would stay bounded but their derivatives would not asymptotically vanish: they should oscillate indefinitely (around a constant or not) and thus, their derivatives should also oscillate around zero without beeing damped. What are the variables able to behave in this way?\
We can reasonably assume that it is not $x_\pm$, otherwise it would mean that the variation of the Hubble constant would be anisotropic. Since $x_\pm$ and $\dot x_\pm$ have to vanish, it implies that $w$ also asymptotically vanishes and consequently, $\dot w\rightarrow 0$. Finally the only variables whose equilibrium is not necessary to isotropy and whose derivatives could be oscillating are $y$ and $z$. Under these assumptions, what about $\ell$ behaviour?\
Equation (\[eq4\]) writes asymptotically $\dot z = y^2(72z-12\ell)$. Since we have assumed that $\dot z$ was oscillating, it shows that $\ell$ can not tend to a constant, diverge monotonically or diverge oscillatory if the oscillations are not large enought($\ell=\Omega^{1/3}+\cos \Omega$ as instance) otherwise the sign of $\dot z$ would be constant. Consequently, only oscillatory $\ell$ with sufficiently large amplitudes and not tending to a constant may allow oscillations of $z$ too ($\ell=\Omega \sin \Omega$ or $\ell=n \cos \Omega$ with $n$ a constant 