Hénon-Heiles System

A 2D nonlinear Hamiltonian — regular orbits vs chaotic trajectories

E = 0.083
12
8

Position space (x, y)

Poincaré section (y=0, ṗ_y > 0)

E = 0.083  |  threshold E_c = 1/6 ≈ 0.1667

The Hénon-Heiles Hamiltonian is H = ½(ẋ²+ẏ²+x²+y²) + x²y − y³/3. Below the critical energy E_c = 1/6 most orbits are regular (smooth KAM tori). Above it, chaos proliferates — orbits explore a larger region ergodically. The Poincaré section (right panel) plots (y, ṗ_y) whenever x=0, revealing whether orbits lie on tori (smooth curves) or are chaotic (scattered points).