Double pendulum.
A double pendulum hanging from a fixed pivot — two rods, two bobs, four coupled equations of motion. Pick the starting angles and release. Then spawn ghost twins, each offset by a tiny ε, and watch them track the primary for a few seconds before chaos pulls them apart. The Lyapunov time is the rough scale on which a perturbation grows by e; after roughly ten of them, the ghosts have forgotten where they came from.
What you're looking at.
The motion is fully deterministic — four coupled ordinary differential equations in θ₁, θ₂, ω₁, ω₂, dropping out of the Lagrangian for two coupled rigid rods. There is no randomness anywhere in the integrator. Two pendulums started at exactly 89.999° and 90.000° look identical for a while and then completely separate; that's the chaos signature in the bare equations.
The Lyapunov time is the timescale over which a tiny perturbation grows by a factor of e ≈ 2.718. For a double pendulum well inside the energetic regime it's a few seconds. After about ten Lyapunov times any positional information about the starting state has effectively been thrown away — the ghosts are wandering an attractor that the primary also lives on, but their phase relative to it is random.
This is not a numerical-error artefact. Even an idealised analog computer with infinitely precise components would show the same divergence, because the divergence is baked into the equations. RK4 with a small step buys you accuracy, not stability against chaos. Crank up the damping slider and the chaos drains away as energy leaks — turn it back to zero and the ghosts fan out again.