= Solution
For $c=1$, the even equilibria are \b[<stable foci>] with <eigenvalues> $(-1\pm i\sqrt3)/2$, and the odd equilibria are \b[<saddle equilibria>] with <eigenvalues> $r_\pm=(-1\pm\sqrt5)/2$. The local saddle directions are $y=r_\pm[x-(2j+1)\pi]$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/ia/paper-2-pendulum-phase.png]
{title=Phase portrait of a pendulum with unit damping, showing spiral sinks and the stable and unstable saddle branches}
The <phase portrait> shows clockwise spiraling into each even equilibrium: on its right-hand horizontal axis, the flow initially points down. The stable <separatrices> of the saddles divide attraction basins on the unwrapped angle axis, while the unstable branches flow into neighboring sinks. Initial conditions with enough <mechanical energy> can cross one or more potential crests before being captured. The curves repeat under $x\mapsto x+2\pi$. They are trajectories, not conservative energy contours: <energy dissipation of a damped pendulum> rules out nonconstant periodic orbits.
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