a function of $\phi$, provided that $\phi-\Omega$ relation is invertible. Here, if $v$ is bounded by a positive constant from below and if $1-v$ is non-negative, one can easily check that $x_\pm$ and $w$ have required asymptotic behavior. Thus, the only possible constraint on $v(\Omega)$ is the invertibility of the $\phi-\Omega$ relation. For example, the choice: $$v=v_0+\frac{1}{\Omega}\sin^2\Omega^3$$ satisfies this condition, if $v_0$ is a constant in the range $0<v_0<1$. However, for this choice, $\ell$, $\dot y$ and $\dot z$ diverge oscillatorily as $\Omega\rightarrow -\infty$, although the isotropization condition $x_\pm\rightarrow 0$ as $\Omega\rightarrow -\infty$ is satisfied and $(y,z)$ are bounded.

The Bianchi type $VI_0$ and $VII_0$ models {#s12}
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The results are similar to those of Bianchi type $II$ model.

### Field equations {#s121}

We will use the following variables for both Bianchi type $VI_0$ and $VII_0$ models: $$\begin{aligned}
&x_\pm=p_\pm H^{-1}&\\
&y=\pi R_0^3e^{-3\Omega}U^{1/2}H^{-1}&\\
&z=p_\phi\phi (3+2\Omega)^{-1/2}H^{-1}&\\
&w_\pm=\pi R_0^2e^{-2\Omega+2\beta_{+}\pm 2\sqrt{3}\beta_-} H^{-1}\label{var41}&\\\nonumber\end{aligned}$$ They are independent since they respectively depend on $p_\pm$, $\phi$, $p_\phi$ and a combining of $\beta_\pm$. The definitions of $x_\pm$, $y$ and $z$ are the same as these of the Bianchi type $II$ model. The Hamiltonian is written: $$x_+^2+x_-^2+24y^2+12z^2+12(w_+\pm w_-)^2=1\label{contrainte1}$$ and the field equations become: $$\begin{aligned}
&\dot{x}_+=72y^2x_++24(x_+-1)(w_-\pm w_+)^2\label{eq12}&\\
&\dot{x}_-=72y^2x_-+24x_-(w_-\pm w_+)^2+24\sqrt{3}(w_-^2-w_+^2)\label{eq22}&\\
&\dot{y}=y(6\ell z+72y^2-3+24(w_-\pm 