How does a smooth Hamiltonian system slide from order into chaos as its energy rises?
▶ Launch the interactive simulationIntegrate orbits of the Hénon–Heiles galactic potential and take a Poincaré section (their crossings of the plane x=0); a regular orbit pierces it on a smooth closed curve, a chaotic one scatters. Each orbit's Lyapunov exponent colours it blue (regular) or warm (chaotic).
the KAM scenario: at low energy nested invariant tori (regular), which progressively break into a chaotic sea as the energy climbs toward the escape energy 1/6