Chaos

A dynamical system is chaotic when two trajectories starting from arbitrarily close initial conditions separate exponentially fast: |δx(t)| ≈ |δx(0)| · e^(λt), where λ > 0 is the largest Lyapunov exponent. The system is fully deterministic — no randomness — but the exponential amplification of initial-condition error makes long-term prediction impossible in practice.

Chaos is common in Hamiltonian systems, typically associated with the destruction of invariant tori that KAM theorem protects only under small perturbations. The double pendulum, the three-body problem, and the driven damped pendulum are canonical chaotic systems. Weather prediction is bounded by the roughly 2-week Lyapunov time of the atmosphere.

Discovered by Henri Poincaré in 1890 in his study of the three-body problem; rediscovered by Edward Lorenz in 1963 in a simple model of atmospheric convection.