Double pendulum and chaos
A hair's breadth at the start, worlds apart in the end.
Hang a second pendulum from the first. The equations of motion are completely deterministic, yet long-term prediction is impossible: two double pendulums whose initial angles differ by only 0.1° follow completely different paths after a few seconds, and the difference grows roughly exponentially. This is chaos. At small angles the motion is close to a superposition of two normal modes, and the difference does not grow.
Tip separation (log)
Log vertical axis: a roughly straight rising line means exponential growth; at small angles there is almost no growth.
Think about it
If both initial angles are set within 20°, does the control pendulum's deviation still grow rapidly?
Hardly. At small angles the double pendulum is approximately a linear system: its motion is a superposition of two normal modes, and the deviation changes only slowly with time. Chaos appears only at larger angles, where nonlinearity is significant.
Model assumptionsBoth arms 1 m, bobs 1 kg · Friction and air drag ignored · RK4 integration (step 2 ms) · Green control pendulum's θ₂ is larger by δ
Data log
Set the parameters and click “Record data”. Record several rows with different parameters to plot their relationship and fit a straight line; pause and record at different times to study how a reading changes with time.