The system is periodical; not counting the absolute angle but the velocities, angular velocities, and the angle at the joint are all periodical functions.

When the angle at the joint is 180° the Energy and Angular momentum determines v_1 and v_2, the velocities of the masses. It is known that Energy and Angular momentum both conserve.

Therefore the system will play out the same after states when the inner joint is 180°, and all the parameters will be periodical between 2 such states.

You can observe this in the demonstration in TFA: set gravity to 0, and observe how the graph rotates, especially the furthest points from the origin (where the angle at the joint is 180°).

edit: there are probably 2 different solutions for when the joint angle is 180°? I've found a graph where one furthest place from the origin is really pointy, and then the other is rather round.