Let be
$$ \left\{ \begin{array}{lcc} \dot{x_1}=-ax_1^2+bx_1x_2+cx_2^2 \\ \\ \dot{x_2}=-x_2+dx_1^2+mx_1x_2+nx_2^2 \end{array} \right.$$ a system; $a,b,c,d,m,n\in \mathbb{R}$ constants.
$1.$ If $a\neq0$ then $\bar{x}=0$ is inestable.
$2.$ If $a=0$ and $bd<0$ then $\bar{x}=0$ is asymptotically stable.
$3.$ If $a=0$ and $bd>0$ then $\bar{x}=0$ is unestable.
$4.$ If $a=b=0$ and $cd\ne0$ then $\bar{x}=0$ is unestable.
$5.$ If $a=b=c=0$ then $\bar{x}=0$ is stable but not asymptotically stable.
$6.$ If $a=d=0$ then $\bar{x}=0$ is stable but not asymptotically stable.
Let be $f_1(x_1, x_2)=-ax_1^2+bx_1x_2+cx_2^2$ and $f_2(x_1, x_2)=dx_1^2+mx_1x_2+nx_2^2.$ The idea is to find a function $\phi(x_1)$ such that $-\phi(x_1)+f_2(x_1, \phi(x_1))=0$ and $\phi(0)=0,\phi'(0)=0$ and apply the following result:
Theorem: Let's suppose that $f_1(x_1, \phi(x_1))=px_1^k+\Theta(|x_1|^{k+1})$ when $x_1\to 0$,$p\ne 0$ and $k\in \mathbb{N}.$
Then the equilibrium $\bar{x}=0$ of system is asymptotically stable if $p<0$ and $k$ odd and unstable otherwise.