whatA kicked rotor stripped to two lines of arithmetic: momentum gains K sin theta, angle gains momentum. Phase space fills orbit by orbit -- each initial condition either traces a closed curve of order or splashes into the chaotic sea. Click anywhere to launch your own.

whyThe claim: as the kick K grows, the KAM islands shrink in a rising chaotic sea -- and this is measured, not asserted: a tangent-map Lyapunov classifier sweeps a 24 by 24 grid of initial conditions and reports the sea fraction live. The second measurement is the fence: below K near 0.97 the last invariant curves cage a chaotic bundle's momentum (watch delta-p saturate); above, the cage breaks and momentum leaks without bound. Order and chaos interleaved at every scale -- islands inside chains inside islands.

howSeeded initial conditions deposit into a density grid, ink accumulating like an etching plate. The Lyapunov classifier calls an orbit chaotic when its finite-time exponent clears 0.05 over 1200 kicks -- a declared threshold, not a truth. The separatrix bundle starts by the hyperbolic point at theta = 0 with momentum unwrapped so leakage can be seen whole. Moving the slider honestly clears the plate and remeasures.