the Lyapunov exponent
/ lee-ah-POO-nof /
Sensitive dependence is a yes-or-no story — do nearby trajectories fly apart or not? The Lyapunov exponent turns it into a number: a single rate that measures exactly how fast two infinitesimally close starts separate. It is the dial that reads positive for chaos, zero for borderline, and negative for stable, and its size tells you how quickly prediction goes blind.
Take two trajectories that start a tiny distance d_0 apart and let d(t) be their separation. The Lyapunov exponent lambda is defined so that, on average, d(t) grows like d_0 e^(lambda t). Solving for lambda, it is the long-time average growth rate of the logarithm of the separation: lambda = limit as t -> infinity of (1/t) ln( d(t) / d_0 ), averaged over the attractor. The sign is everything. If lambda < 0, nearby trajectories converge — the system is settling onto a stable fixed point or cycle, and prediction is easy. If lambda = 0, separations stay roughly constant, the marginal case of, say, neutral oscillations. If lambda > 0, separations blow up exponentially — that positive exponent is the precise, quantitative fingerprint of chaos. (In d dimensions there is a whole spectrum of d exponents, one per direction; the largest one decides chaos.)
The Lyapunov exponent is the standard diagnostic that turns 'this looks chaotic' into a measurable, reproducible verdict, and its reciprocal 1/lambda sets a natural prediction horizon — the timescale over which a small uncertainty grows to dominate. It is also the honest reason chaos limits forecasting in a calculable way: knowing lambda, you can estimate how much extra initial precision buys how much extra prediction time, and the answer is always 'painfully little'. A positive Lyapunov exponent, together with bounded motion, is one of the cleanest working definitions of deterministic chaos.
For the logistic map at r = 4, the Lyapunov exponent works out to lambda = ln 2 per step (about 0.693), positive — so separations double every iteration on average, confirming chaos. For r = 2.5, where iterates settle to a fixed point, lambda is negative, confirming nearby starts converge and the dynamics is predictable.
A positive exponent (here ln 2) certifies chaos; a negative one certifies convergence to a stable state.
The exponent is an average over the whole attractor, not a local rate: along a chaotic orbit the instantaneous stretching speeds up and slows down, sometimes even contracting, and only the long-run average is positive. A single short stretch of fast separation does not by itself prove a positive Lyapunov exponent.