Explain what is meant by a strict Lyapunov function on a domain $\mathcal{D}$ containing the origin for a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$. Define the domain of stability of a fixed point $\mathbf{x}_{0}$.

By considering the function $V=\frac{1}{2}\left(x^{2}+y^{2}\right)$ show that the origin is an asymptotically stable fixed point of

$$\begin{aligned} &\dot{x}=-2 x+y+x^{3}-x y^{2} \\ &\dot{y}=-x-2 y+6 x^{2} y+4 y^{3} \end{aligned}$$

Show also that its domain of stability includes $x^{2}+y^{2}<\frac{1}{2}$ and is contained in $x^{2}+y^{2} \leqslant 2$.