Hamiltonian Chaos

Double pendulum: sensitivity to initial conditions & Poincaré sections

Phase Portrait / Poincaré
Double Pendulum
A double pendulum is a canonical Hamiltonian system: energy is conserved but the motion is chaotic for large amplitudes. Nearby trajectories diverge exponentially (positive Lyapunov exponent). The Poincaré section (left) shows the intersection of orbits with a plane in phase space — regular orbits form closed curves (KAM tori), chaotic orbits fill areas (Arnold diffusion).