System A
System B
How different System B's starting angle 1 is from System A's, in degrees
How a tiny difference turns into total unpredictability
A double pendulum — one rod swinging from another — is a shockingly chaotic system: nothing about its equations is random, yet its future is nearly impossible to predict. Below, two identical double pendulums start almost exactly together, differing only by a fraction of a degree in one starting angle. Watch their paths peel apart.
Both pendulums obey the exact same deterministic equations, with the exact same masses, lengths, and gravity — the only difference between them is the tiny starting-angle gap set by Initial angle difference below. For a while they swing together almost perfectly. Then, seemingly out of nowhere, their outer bobs (the trailing dots) start tracing completely different paths.
This is the hallmark of a chaotic system: not that it's random, but that arbitrarily small differences in starting conditions grow exponentially over time, making long-term prediction practically impossible even though every step of the motion is fully determined. Try the Gentle preset for contrast — at small angles, a double pendulum barely diverges at all.