KAM tori · chaotic sea · Chirikov standard map · phase-space structure
System: Hénon-Heiles
Points: 0
E_esc = 1/6 ≈ 0.167
Poincaré section — For the 2-DOF Hénon-Heiles Hamiltonian
H = (p₁²+p₂²)/2 + (q₁²+q₂²)/2 + q₁²q₂ − q₂³/3,
we fix q₂=0 and record (q₁,p₁) whenever q₂ passes through 0 upward.
This reduces 4D phase space → 2D area-preserving map.
KAM theorem (Kolmogorov 1954, Arnold 1963, Moser 1962): most
invariant tori survive small perturbations — they appear as smooth closed curves.
As energy E increases toward the escape value 1/6≈0.167, tori break and
the chaotic sea expands (Birkhoff-Smale horseshoes, heteroclinic tangles).
Standard map (Chirikov 1969): θₙ₊₁ = θₙ + pₙ₊₁, pₙ₊₁ = pₙ + K sin(θₙ).
The KAM tori (last surviving = "golden mean torus") break at the
Chirikov criterion K≈0.972. For K>0.972, global transport is possible
(Arnold diffusion in higher dimensions).