H}{\partial \phi}=-12\frac{\phi p_\phi ^2}{(3+2\omega)H}+12\frac{\omega_\phi \phi^2 p_\phi ^2 }{(3+2\omega)^2 H}-12\pi^2 R_0 ^6 \frac{e^{-6\Omega}U_\phi }{H}$$ $$\label{hamp}
\dot{H}=\frac{dH}{d\Omega}=\frac{\partial H}{\partial \Omega}=-72\pi^2 R_0 ^6 \frac{e^{-6\Omega}U}{H}+\frac{\partial V(\Omega,\beta_+,\beta_-)/\partial \Omega}{2H}$$ We set $N_i=0$ and calculate that, whatever the Bianchi model, the lapse function is given by: $$\label{lapse}
N=\frac{12\pi R_0^3e^{-3\Omega}}{H}$$ Then, the relation between the time $\Omega$ and the proper time $t$ is $dt=-Nd\Omega$.\
Before starting the analysis of each Bianchi model, let us talk about some necessary conditions for isotropisation. By definition, the Universe isotropises if each metric function tends toward a common form, let us say $R^2$. From Misner parameterisation, we deduce that it implies $e^{-\Omega}\rightarrow R$ and $\beta_\pm\rightarrow 0$. A convenient measure of anisotropy is given by the a mesure of the root mean square anisotropy[@RyaShe75] $d\beta_+/dt^2+d\beta_-/dt^2=(d\beta_+/d\Omega^2+d\beta_-/d\Omega^2)(d\Omega/dt)^2$ which have to decay such that isotropy occurs. Assuming that isotropisation is an asymptotic phenomenon arising when proper time diverges, it means that asymptotically $d\beta_\pm/dt\rightarrow 0$. It is this last necessary condition only that we will use in this work. It is not sufficient for isotropisation since it does not prevent $\beta_\pm$ to diverge or to tend toward a big constant but it is necessary if we want that the Hubble factors $H_i=\frac{dg_{ii}}{dt} g_{ii}^{-1}$ be asymptotically the same, i.e. 