(a) A dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ has a fixed point at the origin. Define the terms asymptotic stability, Lyapunov function and domain of stability of the fixed point $\mathbf{x}=\mathbf{0}$. State and prove Lyapunov's first theorem and state (without proof) La Salle's invariance principle.

(b) Consider the system

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

(i) Show that trajectories cannot leave the square $S=\{(x, y):|x|<1,|y|<1\}$. Show also that there are no fixed points in $S$ other than the origin. Is this enough to deduce that $S$ is in the domain of stability of the origin?

(ii) Construct a Lyapunov function of the form $V=x^{2} / 2+g(y)$. Deduce that the origin is asymptotically stable.

(iii) Find the largest rectangle of the form $|x|<x_{0},|y|<y_{0}$ on which $V$ is a strict Lyapunov function. Is this enough to deduce that this region is in the domain of stability of the origin?

(iv) Purely from using the Lyapunov function $V$, what is the most that can be deduced about the domain of stability of the origin?