Determine Lyapunov stability of an ODE

Question

$$ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t}=-x-3y+2z+yz,\\ \frac{\mathrm{d}y}{\mathrm{d}t}=3x-y-z+xz,\\ \frac{\mathrm{d}z}{\mathrm{d}t}=-2x+y-z+xy.\\ \end{cases} $$ Let Lyapunov function $\mathcal{V}(x,y,z)=x^2+y^2+z^2$, we get $$ \dot{\mathcal{V}}\left( x,y,z \right) =2x\left( -x-3y+2z+yz \right) +2y\left( 3x-y-z+xz \right) +2z\left( -2x+y-z+xy \right) =-2\left( x^2+y^2+z^2-3xyz \right). $$ But here $\dot{\mathcal{V}}$ is neither positively definite nor negatively definite, then how to use Lyapunov's theory to determine its stability? Thank you in advance for your help and contribution!

Answer 1

I've realized that the question does not necessarily require me to determine the stability upon the whole space, but a neighbourhood of zero. Then it's easy to see that $\dot{\mathcal{V}}(x,y,z)$ is negetively definite in the neighbourhood$\{x^2+y^2+z^2<1\}$, so it enjoys asymptotic stability.