Uploaded January 2021 | Updated September 2026, 2 weeks ago
Simulation of 500 double pendulums having initial start positions ranging between 2.2995 radians and 2.3 radians spanned from the vertical line. That is, the difference in starting angle between each adjacent pair of double pendulums is one millionth of a radian. Hence, the outermost apices of all the double pendulums sweeps only 1/2000 of a radian at the very start.
Despite such a tiny difference between the 500 initial conditions, the outermost apices of the pendulums rapidly spread out homogeneously all over their energetically allowed range, as the double pendulum is a chaotic system for certain criteria of the initial conditions, meaning that it is ridiculously sensitive to initial conditions. This is sometimes referred to as the "Butterfly effect".
CPU time was ~3 minutes for calculating all positions of the pendulums for all 3200 frames on a mid-2012 MacBook Air. Plotting/saving all frames took ~1 hour in total.
Here is a good example page to start out playing with one double pendulum in Python: scipython.com/blog/the-double-pendulum/. My scripts need a thorough clean-up before sharing.
Music by @gpcbass!
Done in Python and ffmpeg.
Simulation of 500 double pendulums having initial start positions ranging between 2.2995 radians and 2.3 radians spanned from the vertical line. That is, the difference in starting angle between each adjacent pair of double pendulums is one millionth of a radian. Hence, the outermost apices of all the double pendulums sweeps only 1/2000 of a radian at the very start.
Despite such a tiny difference between the 500 initial conditions, the outermost apices of the pendulums rapidly spread out homogeneously all over their energetically allowed range, as the double pendulum is a chaotic system for certain criteria of the initial conditions, meaning that it is ridiculously sensitive to initial conditions. This is sometimes referred to as the "Butterfly effect".
CPU time was ~3 minutes for calculating all positions of the pendulums for all 3200 frames on a mid-2012 MacBook Air. Plotting/saving all frames took ~1 hour in total.
Here is a good example page to start out playing with one double pendulum in Python: scipython.com/blog/the-double-pendulum/. My scripts need a thorough clean-up before sharing.
Music by @gpcbass!
Done in Python and ffmpeg.
