Undamped simple pendulum:
      We plot the phase portrait of undamped ideal pendulum with Chebfun:


We see that for small enough initial velocities, the pendulum swings back and forth around the equilibrium u=0, while for larger initial velocities, it swings over and over the top position. However, if we introduce damping, all trajectories will eventually end up at rest:
Phas portrait for the damped pendulum, plotted with Chebfun.