Periodic solutions of the double pendulum

Question

I'm stuck: Are there periodic solutions of the double pendulum, or not?

The question is four-fold:

1. Does every double pendulum - defined by two masses $m_1$, $m_2$ and lengths $l_1$, $l_2$ - have periodic solutions (in phase space)? (What's for sure: If there is a solution, it's highly instable.)

2. If not so: Does any double pendulum have periodic solutions? If so: How must $m_1$, $m_2$, $l_1$, $l_2$ relate?

3. If so: Can initial conditions which give rise to periodic solutions be explictly given? (If so: an example?)

4. Is there a chance to calculate and visualize a periodic solution? (I've never seen one to this day.)

Might it be the case that for each solution that looks like a periodic solution at first glance, there is a solution that is really periodic, e.g. for this one:

Which in fact is not periodic (at least not obviously):

Answer 1

For clarity, I refer to the following definitions of the double pendulum parameters, and refer to the equations of motion listed here.

Answering in order of questions asked:

1. Although I am not sure if every double pendulum has a mathematically periodic solution, they can all have at least approximately periodic solutions when the angles $\theta_1$, $\theta_2$ and their rates of change $\dot{\theta_1}$, $\dot{\theta_2}$ are small. This is because the only nonlinear terms in the equations of motion involve these angles, and in the small-angle limit one can accurately linearize the equations of motion and find normal modes which are approximately normal modes of the true system.
2. Yes—namely, if you take $l_1 = l_2$, and make the initial angles and their rates of change the same ($\theta_1 =\theta_2 = \theta$, $\dot{\theta_1} = \dot{\theta_2} = \dot{\theta}$), this will generate a periodic motion where the double pendulum swings as one single pendulum. This is entirely independent of the masses, and can be verified from direct substitution into the equations of motion since it causes the evolution equations for each angle to both decouple and become identical.
3. See above.
4. Although the solution mentioned above is certainly periodic, it is also unstable. Any slight difference in the angles will cause the nonlinearities to emerge and grow, throwing off the periodicity. Thus, modeling a periodic solution for a double pendulum numerically from the base equations is nearly impossible, which is probably why you haven't seen this. I would wager this is also why the solution you demonstrate appears periodic but fails to be in the long-term; you've likely hit on a periodic solution (or very close to it), but you are either not at the exact initial conditions needed or numerical error causes the periodicity to vanish eventually.