(a) State Liapunov's first theorem and La Salle's invariance principle. Use these results to show that the fixed point at the origin of the system

$$\ddot{x}+k \dot{x}+\sin ^{3} x=0, \quad k>0,$$

is asymptotically stable.

(b) State the Poincaré-Bendixson theorem. Show that the forced damped pendulum

$$\dot{\theta}=p, \quad \dot{p}=-k p-\sin \theta+F, \quad k>0,$$

with $F>1$, has a periodic orbit that encircles the cylindrical phase space $(\theta, p) \in \mathbb{R}[\bmod 2 \pi] \times \mathbb{R}$, and that it is unique.

[You may assume that the Poincaré-Bendixson theorem also holds on a cylinder, and comment, without proof, on the use of any other standard results.]

(c) Now consider $(*)$ for $F, k=O(\epsilon)$, where $\epsilon \ll 1$. Use the energy-balance method to show that there is a homoclinic orbit in $p \geqslant 0$ if $F=F_{h}(k)$, where $F_{h} \approx 4 k / \pi>0$.

Explain briefly why there is no homoclinic orbit in $p \leqslant 0$ for $F>0$.