Show the asymptotical stability at the origin of a non linear system

Question

Show that at the origin the following system has an asymptotically stable point:
$\begin{cases} \dot{x_1}=-\phi_1(x_1)+\phi_2(x_2)\\ \dot{x_2}=\phi_1(x_1)-\phi_2(x_2)\\ \end{cases}$ ,with $x_i^2\alpha_i\leq \phi_i(x_i)x_i\leq x_i^2\beta_i$, $\beta_i>\alpha_i>0$ for $i=1,2$.
I have tried $V(x)=\frac{1}{2}x_1^2+\frac{1}{2}x_2^2$ but in the derivative unfortunately appears the term $x_2\phi_1(x_1)+x_1\phi_1(x_2)$.
Can you help me to find out the right lyapunov function?