I got some trouble with a problem about Lyapunov functions. I want to show that $V(x,y)=x^2+y^2$ is a weak Lyapunov function for the system
$$\begin{cases} x'=y\\ y'=-x-y^3(1-x^2)^2 \end{cases}$$
So my solution so far is that calculation of $$\nabla V \cdot f= -2y^4(1-x^2)^2$$ for which I want to show that $-2y^4(1-x^2)^2 \leq 0$.
However, to me this looks like a strong Lyapunov function, since $(0,0)$ is not a point of interest, and it is positive if we look at every other point, right?
No, it's not a strong Lyapunov function, since $\nabla V \cdot f$ equals zero along the whole line $y=0$ (which implies that there is no punctured neighbourhood of $(0,0)$ where it's less than zero).