State without proof Lyapunov's first theorem, carefully defining all the terms that you use.

Consider the dynamical system

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

By choosing a Lyapunov function $V(x, y)=x^{2}+y^{2}$, prove that the origin is asymptotically stable.

By factorising the expression for $\dot{V}$, or otherwise, show that the basin of attraction of the origin includes the set $V<7 / 4$.