the Lyapunov exponent
/ lyah-POO-nof /
Sensitive dependence is qualitative: nearby trajectories run away from each other. The Lyapunov exponent is the number that makes it quantitative: it measures exactly how fast they run. A positive value is the mathematical certificate of chaos; a negative one says nearby states are being pulled together instead.
Take two trajectories separated initially by delta_0 and follow their separation delta(t). The (largest) Lyapunov exponent is the long-time average exponential rate of that separation: lambda = lim (as t -> infinity, delta_0 -> 0) of (1/t) ln( delta(t) / delta_0 ). Equivalently delta(t) ~ delta_0 e^(lambda t). If lambda > 0 nearby trajectories diverge exponentially, which is chaos; if lambda < 0 they converge, giving a stable fixed point or limit cycle; if lambda = 0 they separate only polynomially, as at the edge of chaos or along the flow direction. In an N-dimensional system there is a whole spectrum of N exponents, one per direction, whose sum gives the average rate of contraction of phase-space volume; for a dissipative system that sum is negative even when the largest exponent is positive.
The Lyapunov exponent is the single most useful diagnostic of chaos, and its reciprocal, the Lyapunov time 1/lambda, sets the predictability horizon, the timescale over which forecasts remain meaningful (about days for weather, tens of millions of years for the inner planets' orbits). Two honest caveats: it is an average over the attractor, so the local divergence rate fluctuates from place to place; and a positive largest exponent alone does not by itself prove chaos unless the motion is also bounded and non-periodic, since an ordinary unstable fixed point also has a positive exponent locally.
The Lorenz attractor has a largest Lyapunov exponent of about 0.9 per unit time, giving a Lyapunov time near 1.1, so an initial uncertainty grows by a factor of e roughly every 1.1 time units, and by a factor of a thousand in about eight.
One number turns unpredictable into a schedule for how fast prediction decays.
A positive Lyapunov exponent certifies exponential stretching but not chaos by itself: the motion must also be bounded and aperiodic, since an isolated unstable fixed point stretches nearby trajectories too yet is not chaotic.