(i) State and prove Lyapunov's First Theorem, and state (without proof) La Salle's Invariance Principle. Show by example how the latter result can be used to prove asymptotic stability of a fixed point even when a strict Lyapunov function does not exist.

(ii) Consider the system

$$\begin{aligned} &\dot{x}=-x+2 y+x^{3}+2 x^{2} y+2 x y^{2}+2 y^{3}, \\ &\dot{y}=-y-x-2 x^{3}+\frac{1}{2} x^{2} y-3 x y^{2}+y^{3} \end{aligned}$$

Show that the origin is asymptotically stable and that the basin of attraction of the origin includes the region $x^{2}+2 y^{2}<2 / 3$.