Consider the damped pendulum equation

$$\frac{d^{2} \theta}{d t^{2}}+c \frac{d \theta}{d t}+\sin \theta=0$$

where $c$ is a positive constant. The energy $E$, which is the sum of the kinetic energy and the potential energy, is defined by

$$E(t)=\frac{1}{2}\left(\frac{d \theta}{d t}\right)^{2}+1-\cos \theta$$

(i) Verify that $E(t)$ is a decreasing function.

(ii) Assuming that $\theta$ is sufficiently small, so that terms of order $\theta^{3}$ can be neglected, find an approximation for the general solution of $(*)$ in terms of two arbitrary constants. Discuss the dependence of this approximate solution on $c$.

(iii) By rewriting $(*)$ as a system of equations for $x(t)=\theta$ and $y(t)=\dot{\theta}$, find all stationary points of $(*)$ and discuss their nature for all $c$, except $c=2$.

(iv) Draw the phase plane curves for the particular case $c=1$.