The Single Pendulum
Two pendulums start in almost exactly the same spot — just like before. But this time there's only one swinging arm. Can you guess where each one will be in ten seconds?
Try it
Turn the tiny difference all the way up. Do the two pendulums ever split apart the way the double pendulum did?
They stayed together
The faint pendulum started a tiny bit different from the bright one — but it stayed right next to it the whole time. The little difference stayed little.
That means if you know where one pendulum is, you can guess where the other one is too. This swing is predictable.
Most simple things are predictable
Same rules, same tiny difference as the double pendulum — but a totally different result. Here the futures don't fan out. A clock, a swing, the seasons: all predictable.
So why was the double pendulum so wild? Adding just one more joint changed everything. Go back and compare →
For teachers & grown-ups
A single pendulum is governed by θ̈ = −(g/L)·sin θ, integrated here with 4th-order Runge–Kutta — the same solver used for the double pendulum, so the comparison is apples-to-apples. The key difference: this system is not chaotic. Nearby starting conditions stay nearby — their separation stays bounded instead of growing exponentially. (At large amplitudes the period depends slightly on amplitude, so two arms with different swing heights will drift very slowly out of phase over many swings — worth pointing out to older students as "slow and predictable" versus the double pendulum's "fast and not.") This is the control case that makes deterministic chaos meaningful: chaos isn't "complicated rules," it's sensitive dependence on initial conditions.