KAM Theorem: Standard Map Phase Portrait

K = 0.500
N = 30
1000
Chirikov Standard Map: p_{n+1} = p_n + K·sin(θ_n), θ_{n+1} = θ_n + p_{n+1} (mod 2π). This 2D area-preserving map is the paradigm of Hamiltonian chaos. For K=0: integrable (horizontal lines). For 0<K<K_c: KAM tori survive. At K_c≈0.9716: the last KAM torus (golden-mean winding ω=(√5−1)/2) breaks. For K>K_c: global chaos.

KAM Theorem (Kolmogorov 1954, Arnold 1963, Moser 1962): For sufficiently irrational winding numbers (satisfying a Diophantine condition |ω−p/q| > C/q^{2+ε}), KAM tori persist under small perturbations. Rational tori break into resonance islands (Birkhoff fixed points) surrounded by homoclinic chaos.

Last KAM torus at K_c (Greene 1979): winding number ω = (√5−1)/2 (golden mean, most irrational). Highlighted in gold.