(i) Define a Liapounov function for a flow $\phi$ on $\mathbb{R}^{n}$. Explain what it means for a fixed point of the flow to be Liapounov stable. State and prove Liapounov's first stability theorem.

(ii) Consider the damped pendulum

$$\ddot{\theta}+k \dot{\theta}+\sin \theta=0,$$

where $k>0$. Show that there are just two fixed points (considering the phase space as an infinite cylinder), and that one of these is the origin and is Liapounov stable. Show further that the origin is asymptotically stable, and that the the $\omega$-limit set of each point in the phase space is one or other of the two fixed points, justifying your answer carefully.

[You should state carefully any theorems you use in your answer, but you need not prove them.]