The double pendulum is widely known for being chaotic, in stark contrast to the well behaved single pendulum. But can it be analyzed statistically? I tried to get to the bottom of this by doing some simulations to find, whether the pendulum would have a kind of average frequency.
I simulated a double pendulum in python and recorded the displacement of the first bob. I then did a Fourier transform on the displacement. This essentially filters out the most important frequencies of the signal. I found out that for small initial displacements (10° - 20°) the double pendulum is very well behaved, having a main frequency similar to a single pendulum the length of the entire double pendulum and a pretty strong higher frequency 2.4 times higher.
When I started the pendulum with a higher initial displacement, there were more overtones. At a certain point, the pendulum undergoes an abrupt transition. For an initial displacement of 79.141958°, the pendulum still has distinct frequencies, but for 79.141959° there are no discernible frequencies at all.
Imgur album with diagrams: https://imgur.com/a/rpJ0QcJ
PS.: Was for some reason deleted in r/physics, I thought it would fit pretty well here though.
Edit: I tried the simulations with the initial condition being the first rod being dsiplaced to the right, the second rod to the left by the same angle (the Energy for a given angle stays the same as before). This results in a different picture https://imgur.com/a/klkGtg4 (just 100 samples though), kinda the opposite of bifurcation. To get results that are only dependent on input energy, I'd have to do much longer simulations, which aren't really useful with runge kutta. The "transition to chaos" is more gradual but still finishes about 70-80°.
Hi everyone, I would like to share this simulation of the Double Pendulum
Link: https://www.geogebra.org/m/yjz98xay
I used the equations form MyPhysicsLab:
https://www.myphysicslab.com/pendulum/double-pendulum-en.html
