Chirikov standard map: transition from integrable to chaos as kick strength K increases
Regime: KAM tori mostly intact
Standard map: p' = p + K·sin(q), q' = q + p' (mod 2π)
K = 0: integrable (horizontal lines). K ≈ 0.97: last KAM torus (golden mean) breaks (Chirikov 1979).
K > 1: chaotic sea dominates. Islands of stability persist at all K. Quantum version shows
dynamical localization — diffusion halts after a quantum break time.