Let $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ be a two-dimensional dynamical system with a fixed point at $\mathbf{x}=\mathbf{0}$. Define a Lyapunov function $V(\mathbf{x})$ and explain what it means for $\mathbf{x}=\mathbf{0}$ to be Lyapunov stable.

Determine the values of $\beta$ for which $V=x^{2}+\beta y^{2}$ is a Lyapunov function in a sufficiently small neighbourhood of the origin for the system

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

What can be deduced about the basin of attraction of the origin using $V$ when $\beta=2 ?$