Chirikov Standard Map — KAM Tori & Chaos

The Chirikov standard map is the paradigmatic area-preserving map, modeling kicked rotors and many Hamiltonian systems. KAM tori persist for small k but break at k ≈ 0.9716 (Greene's criterion). Above the critical k, global chaos with Arnold diffusion emerges.

Parameters

k=0.97 (near critical)
p_{n+1} = p_n + k·sin(θ_n) (mod 2π)
θ_{n+1} = θ_n + p_{n+1} (mod 2π)
Critical k_c ≈ 0.9716 (Greene)
KAM tori: irrational winding ω
Golden mean ω = (√5-1)/2
last KAM torus breaks at k_c
Lyapunov λ > 0 in chaotic sea