Chaos theory: Lorenz attractor and Lyapunov exponents

In 1963 Edward Lorenz ran a weather simulation on an early Royal McBee computer and re-entered a result rounded to 0.506 instead of 0.506127. The output diverged completely. This chance discovery -- initial difference of 0.000127 -- launched chaos theory and redefined the limits of predictability.

• Weather forecasting uses ensemble methods (50-100 perturbed runs) to cope with Lorenz-type chaos -- directly applying Lyapunov exponent theory.
• SpaceX trajectory computation accounts for chaotic sensitivity in upper atmosphere by running Monte Carlo ensembles with bounded Lyapunov estimates.

Lorenz attractor

In 1963 Edward Lorenz discovered that a system of three ODEs modeling atmospheric convection exhibits sensitive dependence on initial conditions: two trajectories starting 0.000001 apart diverge exponentially within hours. This butterfly effect -- a term Lorenz coined in 1972 -- showed that long-term weather prediction is fundamentally limited. The Lorenz attractor is a fractal set with Hausdorff dimension ~2.06.

Lyapunov exponents spectrum

A d-dimensional system has d Lyapunov exponents lambda_1 >= lambda_2 >= ... >= lambda_d. For the Lorenz system: lambda_1 approx +0.9, lambda_2 = 0, lambda_3 approx -14.6. The sum equals the divergence of the vector field (Liouville's theorem): for Lorenz sum = -(sigma + 1 + beta) = -13.67. The Kaplan-Yorke dimension DKY = 2 + lambda_1/|lambda_3| approx 2.06.

Key results

• Lorenz system: sigma=10, rho=28, beta=8/3 gives deterministic chaos.
• Positive MLE lambda_1 > 0 = exponential divergence = chaos.
• Liouville: sum of all Lyapunov exponents = div(F).
• Kaplan-Yorke dimension DKY = 2 + lambda_1/|lambda_3| approx 2.06 for Lorenz.