Chirikov Standard Map
KAM tori, resonance islands, and the onset of chaos
p_{n+1} = p_n + K sin(θ_n) (mod 2π)
θ_{n+1} = θ_n + p_{n+1} (mod 2π)
K < K_c ≈ 0.9716… → KAM tori survive (integrable islands). K > K_c → global chaos (Chirikov 1979). Resonances at p = 2πm/n form island chains (rational tori). Irrational tori persist longest (KAM theorem, Arnold-Moser).
K = 0.97 |
Near critical (K_c ≈ 0.9716)
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