Contents of the Maple 10 file "pendulum.mws":

ENGI 9420:   Phase portrait for the simple damped pendulum :   
[Note that in the phase portraits below, the green orbit corresponds to a pendulum with just enough total energy to pass through the upsidedown position in the absence of any damping.   The dark orbit has just slightly less energy and never quite reaches the upsidedown position.   The blue orbit has negative total energy and never gets closer than 60° to the upsidedown position in the absence of any damping.
When µ > 2w, oscillations cease and the stable equilibrium becomes a node.]

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>	with(DEtools):
>	phaseportrait([diff(x(t),t) = y(t), diff(y(t),t) = 0*y(t) - 4*sin(x(t))], [x(t), y(t)], t=-10..20, [[x(0)=-2, y(0)=0], [x(0)=3, y(0)=0], [x(0)=3.15, y(0)=0.1]], x=-4..7, y=-7..7, stepsize=0.1, linecolour=[blue, violet, green], title=`Undamped Pendulum, mu=0, omega=2`);



>	phaseportrait([diff(x(t),t) = y(t), diff(y(t),t) = -0.2*y(t) - 4*sin(x(t))], [x(t), y(t)], t=-10..20, [[x(0)=-3, y(0)=0], [x(0)=3, y(0)=0], [x(0)=3.15, y(0)=0.1]], x=-4..7, y=-7..7, stepsize=0.1, linecolour=[blue, violet, green], title=`Lightly Damped Pendulum, mu=0.2, omega=2`);



>	phaseportrait([diff(x(t),t) = y(t), diff(y(t),t) = -1*y(t) - 4*sin(x(t))], [x(t), y(t)], t=-10..10, [[x(0)=-3, y(0)=0], [x(0)=3, y(0)=0], [x(0)=3.15, y(0)=0.1]], x=-4..7, y=-7..7, stepsize=0.1, linecolour=[blue, violet, green], title=`Damped Pendulum, mu=1, omega=2`);



>	phaseportrait([diff(x(t),t) = y(t), diff(y(t),t) = -5*y(t) - 16*sin(x(t))], [x(t), y(t)], t=-10..10, [[x(0)=-3, y(0)=0], [x(0)=3, y(0)=0], [x(0)=3.15, y(0)=0.1]], x=-4..7, y=-7..7, stepsize=0.1, linecolour=[blue, violet, green], title=`Heavily Damped Pendulum, mu=5, omega=4`);



>

phaseportrait([diff(x(t),t) = y(t), diff(y(t),t) = -20*y(t) - 4*sin(x(t))], [x(t), y(t)], t=-5..30, [[x(0)=-2, y(0)=0], [x(0)=3, y(0)=0], [x(0)=3.15, y(0)=0.1]], x=-3.5..3.5, y=-0.5..0.5, stepsize=0.01, linecolour=[blue, violet, green], title=`Heavily Damped Pendulum, (non-oscillating), mu=20, omega=2`);