Here are some interesting graphics from the Maple software we used:
Solution plot of the equation
Solution plot of
This is a plot of the path of a pendulum in phase space.  X(t) is the displacement, and y(t) is the velocity, x'(t).
This is a damped pendulum.  the Time variable is also shown, but advances to the left, so that the purple end of the curve shows the slowing down.

It is a true pendulum (not linearized); the equations are:
Note that we have -sin(x) rather than -x.
A projection into the x-w-z subspace of the solution of the 4-dimensional differential equation system