Stable singular point using Lyapunov function

Question

Suppose we have the following system of ODEs: $$\begin{pmatrix}x'(t)\\y'(t)\end{pmatrix}=\begin{pmatrix}y(t)-x(t)^3\\-x(t)^3\end{pmatrix}$$ We know that the point $(0,0)$ is a singular point of this system. We are asked the following:

Using an appropriate Lyapunov function, show that $(0,0)$ is a stable singular point.

I tried using a "quadratic-like" function, but I don't think I'm on the right path. Any help would be greatly appreciated!

Answer 1

Hint: Try the Lyapunov $$V=\frac{1}{4}x^4+\frac{1}{2}y^2$$