sensitive dependence on initial conditions

Two weather forecasts start from very nearly identical measurements, differing only in the last decimal place, as if one included the flap of a single butterfly's wings and the other did not. For a while their predictions agree, then they diverge, and within a couple of weeks they describe entirely different weather. This runaway amplification of tiny differences is sensitive dependence on initial conditions, popularly the butterfly effect.

Formally, a system shows sensitive dependence if trajectories that start arbitrarily close together separate at an exponential rate. If two initial states differ by a tiny separation delta_0, their separation at time t grows roughly like delta(t) ~ delta_0 e^(lambda t), where lambda is a positive Lyapunov exponent. Because the growth is exponential, the time over which a prediction stays accurate grows only logarithmically as you improve your knowledge of the initial state: halving the initial error buys only a fixed additional stretch of predictability, not double. Sensitive dependence, together with the dynamics being bounded and non-periodic, is the defining fingerprint of deterministic chaos.

The crucial and often-missed point is that such systems are perfectly deterministic: the same initial condition always yields the same future, with no randomness in the equations. Chaos is not noise. The unpredictability is entirely practical, coming from our inability to specify the initial state with infinite precision, amplified without bound by the dynamics. Edward Lorenz stumbled on this in 1961 when rounding a weather-model input from six digits to three produced a wholly different forecast. This is why long-range weather prediction has a hard horizon no amount of computing power can push past.