Concept

Shadowing

The property that a computed sequence, each step correct to within a small amount, is the true orbit of some nearby starting point. Where it holds, every statistical claim drawn from a computed chaotic orbit is a claim about a genuine orbit, which is what makes such computations trustworthy.

Named by 2 essays across one field — 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 $The orbit a computer draws, and the orbit. Two orbits of the tent map from the same starting fraction plotted against the step number — one computed exactly in whole-number arithmetic and periodic, one computed in double precision and reaching zero.$

The orbit a computer draws

A chaotic orbit computed in floating point is not the orbit of the point it started from. Sometimes it is the true orbit of a nearby point, which is enough; sometimes the arithmetic simply runs out, and the picture is of the rounding.

dynamics · iteration

Named alongside it

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

Iteration Sensitive dependence Binary expansion Chaos Determinism Exact arithmetic Floating point Logistic map Numerical artefact Orbit Periodic orbit Prediction

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