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Concept

Logistic map

The one-parameter rule that sends a number to r times itself times one minus itself, on the unit interval. As the parameter rises it goes from a fixed point through period doubling into chaos, all with one quadratic rule.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

the logistic map at 3.2, iterated from 0.2. A map drawn as a curve with the diagonal across it, and the staircase that iterating it produces.

The staircase that shows the whole orbit

Take a number, feed it to a rule, feed the answer back in. There is a way of drawing that on the rule's own graph which turns the entire future of a starting point into a shape — and the shape is legible.

dynamics · Iteration A fixed point that attracts, and one that does not. The same map at two parameters, with the staircase walking towards the crossing in one and away in the other.

A point that pulls, and a point that pushes

Every crossing of a curve with the diagonal is a value the rule leaves alone. Whether anything ever arrives there is decided by one number — the slope at the crossing — and the picture makes the reason obvious.

dynamics · Fixed points The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points.

The road paved with doublings

Turn one dial slowly and watch what a map settles into. It settles on a point, then on two points, then four, then eight — faster and faster, and the doublings run out at a parameter that is finite.

dynamics · Period-doubling Where the period doubles, and by how much the gaps shrink. The parameters at which the period doubles, with the ratio of consecutive gaps beside them.

A constant that does not care which map

The gaps between successive period doublings shrink by a factor. Measure that factor for the logistic map and you get 4.669. Measure it for a completely different map and you get 4.669, and nobody expected that.

dynamics · Period-doubling 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 Lyapunov exponent, 2.8 to 4. The average rate at which nearby orbits separate, plotted against the parameter.

How fast two orbits part

The word "sensitive" is an adjective. Averaging the logarithm of one derivative along an orbit turns it into a number — one that says how many steps of prediction the map allows, and whose sign says whether it allows any.

dynamics · Sensitive dependence The tent map and the logistic map, joined by a change of coordinate. Two cobweb diagrams side by side — the tent map at slope two and the logistic map at four — with the orbit of one carried to the orbit of the other by a curve drawn between them.

The same map in different coordinates

The tent map and the logistic map at four look nothing alike and are the same map, carried onto each other by a change of variable. Everything either one does the other does, and the change of variable is a sine squared.

dynamics · Iteration

Named alongside it

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

IterationOrbitBifurcationChaosPeriodic orbitSelf-similarityAttractorCobwebConvergenceDerivativeFixed pointLyapunov exponent

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