Butterfly effect

The butterfly effect is the sensitivity of some systems to initial conditions, in which tiny starting differences can grow into large divergences over time.

Sensitivity to initial conditions: A property in which nearby starting states lead to increasingly different outcomes over time. This is the general dynamical property captured by the butterfly effect.
Edward Norton Lorenz: An American mathematician and meteorologist whose work helped establish modern chaos theory. His weather-model experiments revealed the sensitivity that inspired the butterfly effect.
Randomness: Unpredictability arising from outcomes that are not fixed by prior states and rules. The butterfly effect can produce unpredictability even when the underlying dynamics are deterministic.
Weather forecasting: The prediction of atmospheric conditions using observations, physical laws, and computational models. Finite measurement precision limits how far ahead detailed forecasts remain reliable.
Lyapunov exponent: A quantity measuring the average exponential rate at which nearby trajectories in a dynamical system separate. A positive value often quantifies the rapid divergence central to the butterfly effect.
Lorenz attractor: A strange attractor associated with the Lorenz system, with a distinctive butterfly-like shape. Its shape helped give the effect its familiar butterfly imagery.
Linear system: A system whose outputs obey linear relations, so scaling inputs scales responses proportionally. Linear systems generally lack the nonlinear amplification associated with the butterfly effect.
Numerical error: The discrepancy introduced when mathematical calculations are approximated with finite precision or discrete methods. In sensitive simulations, tiny rounding errors can grow until results diverge.
Lorenz system: A three-variable system of nonlinear differential equations that exhibits chaotic dynamics. Its trajectories demonstrate how small state differences can grow dramatically.
1963 Lorenz paper: Edward Lorenz’s paper describing deterministic nonperiodic flow in a simplified atmospheric model. It provided an early mathematical account of sensitive dependence in a weather model.

Linked from 4 pages