Recently, I have been confused by the following problem:
Show that$$\frac{dx}{dt}=y^{3}-x^{3}y^{2}$$ $$\frac{dy}{dt}=-x^{3}$$is Lyapunov asymptotic stable at the equilibrium point $(0,0)$.
I can just show that the equilibrium point $(0,0)$ is stable but can not judge whether it is asymptotic stable by constructing the Lyapunov function $V(x,y)=\frac{1}{4}(x^{4}+y^{4})$. Differentiating the Lyapunov function $V$, we have $\frac{dV}{dt}=-x^{6}y^{2}\leq0$ when $(x,y)\neq(0,0)$. If we want to prove the system is Lyapunov asymptotic stable, we should construct a Lyapunov function whose derivative about $t$ is strictly smaller than $0$ when $(x,y)\neq(0,0)$. What should I do next?
Any suggestions are wellcome! thank you!