Finding a Lyapunov function and proving the stability

Question

I'm trying to find a Lyapunov function for $(0, 0)$ in the system

\begin{cases} x' = 2x - 2y - (2x - y)^3\\ y' = 4x - 2y + (x - y)^3 \end{cases}

I thought that the following one would appropriate $$V = 2x^2 - 2xy + y^2 = x^2 + (x - y)^2 > 0$$ $$V(0, 0) = 0$$ but is it true that $$V' = (2x + 2x - 2y)(2x - 2y - (2x - y)^3) - 2(x - y)(4x - 2y + (x - y)^3) = -34x^4 - 4y^4 - 60x^2y^2 + 24xy(3x^2 + y^2) \le 0 ?$$

Maybe it is better to find Chetaev function and use other theorem... I would be thankful for any help!

Answer 1

Note that:

$$ V' = -2\,(x - y)^4 -2\,(2\,x - y)^4 \le 0 \quad \forall \, (x,\,y) \in \mathbb{R}^2 $$

so yes, $V$ is a Lyapunov function.