Finding a Lyapunov function

Question

I'm trying to find a Lyapunov function in the system

$$\begin{cases} x'=x^2-1 \\ y'=-xy \end{cases} $$

for the points $(-1,2)$ and $(1,2)$.

First, I'm confused, because this points are not singular points of the system. So, is the question wrong?

Well, if that is the case, assume the singularities $(1,0), (-1,0)$

I tried something like $V(x,y)=ax^3+by^2$, but this leads me to $\dot{V}=2ax^4-2ax^2-2bxy^2$, and I can't do anything.

How can I solve this?

Answer 1

Hint.

$$ y' = -\frac{x}{x^2-1} y\Rightarrow \ln y = -\frac 12\ln(x^2-1)+c_1\Rightarrow y^2(x^2-1) = c_2 $$

so

$$ V(x,y) = y^2(x^2-1) - c_2 $$

is a movement integral and also a Lyapounov function for this dynamic system.