Controls
KAM threshold: K* ≈ 0.9716
Below: closed invariant curves
Above: global chaos
Map: θₙ₊₁ = θₙ + pₙ₊₁
pₙ₊₁ = pₙ + K·sin(θₙ)
(mod 2π)
K=0: Integrable — pure rotation, every orbit is periodic or quasi-periodic.
K small: KAM theorem — most invariant tori survive, perturbed slightly.
K≈0.97: Last KAM torus (Greene's criterion) — the "critical" golden-mean orbit breaks.
K large: Fully developed chaos — orbits fill the torus ergodically.
Click on the canvas to seed individual orbits and watch their trajectories.