A dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ has a fixed point at the origin. Define the terms Lyapunov stability, asymptotic stability and Lyapunov function with respect to this fixed point. State and prove Lyapunov's first theorem and state (without proof) La Salle's invariance principle.

(a) Consider the system

$$\begin{aligned} &\dot{x}=y \\ &\dot{y}=-y-x^{3}+x^{5} \end{aligned}$$

Construct a Lyapunov function of the form $V=f(x)+g(y)$. Deduce that the origin is asymptotically stable, explaining your reasoning carefully. Find the greatest value of $y_{0}$ such that use of this Lyapunov function guarantees that the trajectory through $\left(0, y_{0}\right)$ approaches the origin as $t \rightarrow \infty$.

(b) Consider the system

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

Show that the origin is asymptotically stable and that the basin of attraction of the origin includes the region $x^{2}+x y+y^{2}<\frac{1}{4}$.