Every value in this lesson is rounded to four decimal places from two real physim double-pendulum numerical integrations (scenario examples/double-pendulum-chaotic.json, integrator dp45, one run as-shipped and one with its first angle perturbed by 0.0001 rad), not derived from a closed form. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Two nearly identical starts stay together at first
Two real physim double-pendulum runs start 0.0001 meters apart at t = 0.0000 seconds — only the first angle differs, by one ten-thousandth of a radian. By t = 2.0000 seconds the two outer-bob positions are still only 0.0015 meters apart.
Δ(0.0000)=0.0001m→Δ(2.0000)=0.0015m
The same two runs explode apart — this is chaos
By t = 5.0000 seconds the real separation has grown to 0.0471 meters, and by t = 8.0000 seconds it reaches 1.4896 meters — bigger than the pendulum arms themselves. A one-ten-thousandth-radian nudge became a completely different trajectory; this is the sensitive dependence on initial conditions that separates a chaotic system from the contracting fixed-point maps earlier in this book.