K_c ≈ 0.972 (last KAM)
K: 0.97
Orbits plotted: —
K: 0.97
Orbits plotted: —
Standard map (Chirikov):
p_{n+1} = p_n + K sin(θ_n)
θ_{n+1} = θ_n + p_{n+1} (mod 2π)
KAM theorem: for small K, most invariant tori survive perturbation. The last golden-ratio KAM torus breaks at K ≈ 0.972 (Greene 1979).
K=0: integrable (circles). K→∞: full chaos (Lyapunov ~ ln K/2).
p_{n+1} = p_n + K sin(θ_n)
θ_{n+1} = θ_n + p_{n+1} (mod 2π)
KAM theorem: for small K, most invariant tori survive perturbation. The last golden-ratio KAM torus breaks at K ≈ 0.972 (Greene 1979).
K=0: integrable (circles). K→∞: full chaos (Lyapunov ~ ln K/2).