Is every oscillatory motion periodic in the absence of damping? Well, I asked this title question to my teacher, and he told me that yes, it is true. But I couldn't quite understand how is it possible?
He gave me an example of a simple pendulum and told that it is both periodic and oscillatory. But a pendulum may not return to its mean position every time in the same time interval, right?
Am I missing something?
 A: A simple pendulum is periodic but a double pendulum even without friction, is oscillatory but not periodic.
A: To answer this question I think it's useful to introduce phase plot. At each point in time the pendulum has both a position and a velocity. If you plot the position on the x-axis and the velocity on the y-axis you get what's called a phase plot.
Here is an example of a pendulum with damping

You see that friction causes the velocity and amplitude to go to zero over time. Here is another plot without friction.

Important here is that if you start from a certain point the path is already pre determined. Consider a path where you end up in the point you started from. This means you are guaranteed to repeat the exact same path. This means that each cycle will also take the same amount of time. If this happens the motion is periodic. For a friction-less pendulum this happens so it is both oscillatory and periodic. When there is only a little friction the path won't end up where it started but it will come very close. So the period will be about the same.
A: The period of a pendulum at large amplitudes is not the same as at small amplitudes, but in the absence of friction, it still has a fixed period.
