Two real physim numerical integrations of a double pendulum (scenario examples/double-pendulum-chaotic.json, integrator dp45) supply every value on this page, each rounded to four decimal places from solver output — the two runs start only 0.0001 radians apart in their first angle and are otherwise identical. This is not the contracting, predictable behavior of the fixed-point maps earlier in this book: the outer-bob positions track together for the first couple of seconds, then the real separation grows past the length of the pendulum itself. Real double pendulums add friction, joint play, and measurement uncertainty beyond this rigid-rod, point-mass model.
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.