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Concept

Prediction

A statement about a system's future derived from its present state. In a chaotic system its useful lifetime grows only as the logarithm of the precision of the starting measurement, which is why better instruments buy so little.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Two orbits of the logistic map at 3.9, started 0.0001 apart. Two sequences from almost the same starting point, plotted together against the step number.

A difference too small to draw

Two starting points a ten-thousandth apart, under the same rule, with nothing random anywhere. Within forty steps they have nothing in common — and the rule was not doing anything to them that it does not do to everything.

dynamics · Sensitive dependence A closer start buys time and nothing else. The logarithm of the separation between two Lorenz trajectories plotted against time, for three different initial separations. The three curves are straight and parallel over most of their length, with the same fitted slope.

A closer start buys only time

Two trajectories from almost the same place separate exponentially, and the rate does not depend on how close they began. Halving the initial error buys one fixed interval of extra agreement, and no amount of precision buys more than a fixed number of those.

dynamics · Strange attractor Four outputs are enough to find the rule. A row of 12 outputs of a linear generator, with the first 4 marked as given and the rest as predicted. The multiplier and increment recovered from the given ones reproduce every later output exactly.

Four numbers and the rule is yours

A linear generator can be solved. Given a few of its outputs, the multiplier and the increment fall out of two congruences, and every future output is then known exactly — which is a failure of a completely different kind from the lattice defect, and is not detected by any test of how evenly the points are spread.

computation · Pseudorandomness

Named alongside it

The objects these essays reach for when they reach for this one.

ChaosIterationSensitive dependenceCryptographyDeterminismExponentialInverseLinear systemLogarithmLogistic mapLyapunov exponentModular arithmetic

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