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Concept

Doubling map

The map sending x to 2x with the whole part discarded, on the numbers from 0 to 1. It shifts binary digits one place left, which makes its orbits exactly as varied as the digit sequences, and it is the simplest chaotic system that can be analysed completely.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The orbit of 13/32 under doubling, and its word. A cobweb of the doubling map with one orbit drawn, the interval split in half beneath it, and the letter each step contributes written out in order.

The orbit written as a word

Cut the interval in two and record which half each step of an orbit lands in. The orbit becomes an infinite string of two letters, the map becomes the act of deleting the first letter, and questions about trajectories turn into questions about words.

dynamics · Symbolic dynamics How long four orbits of the doubling map spend in each quarter. Grouped bars for four starting points of the doubling map, each group showing the fraction of time the orbit spends in each quarter of the unit interval, against a dashed line at one quarter.

Almost every orbit is fair

Double a number and keep the fractional part, and do it again, and again. Where the orbit goes is written in the number's binary digits, and for almost every starting point it spends exactly a quarter of its time in each quarter of the interval. The proof is a short argument about Fourier coefficients being pushed to infinity — and it leaves room for a set of exceptions with no length at all, which includes every fraction and, for all anyone can prove, every number anyone has ever named.

dynamics · Iteration A periodic point and a wandering one, both near 0.3, parted by step 4. The distance between the orbit of a periodic point and the orbit of a point from a dense orbit, both starting in the same small interval, plotted against the step until the wandering orbit nears the point farthest from the periodic one.

Sensitivity comes free

The standard definition of chaos asks for three things: an orbit that goes everywhere, periodic orbits everywhere, and sensitive dependence on the starting point. The third, the one the word chaos is usually taken to mean, turns out to follow from the other two. A periodic point and a wandering point that start side by side must eventually part, because the wanderer has to visit places the periodic orbit never goes.

dynamics · Sensitive dependence Every residue joined to 2 times itself, on a dial of 199. A circle with 199 equally spaced points and a chord from each point k to the point 2k mod 199, with the 1-cusped curve the chords envelope drawn dashed.

Multiplying every number on the dial at once

Join every residue on a dial to twice itself and the chords draw a heart-shaped curve with one cusp; join each to three times itself and the curve has two. The picture is the whole multiplication map at once, and it holds three facts: the map splits the dial into cycles whose lengths are orders, those cycles on a dial of 2ⁿ − 1 are the binary necklaces of length n, and the curve is the caustic light draws inside a cup.

discrete · Modular arithmetic Newton's method for z² − 1: two half-planes and the line between. The complex plane split down the imaginary axis into two shaded halves, the roots ±1 marked, and five orbits of Newton's method each converging to the root on its own side.

Chaos on the line between two roots

Newton's method for a quadratic is the one case solved completely: every starting point goes to the nearer root, and the plane splits into two half-planes. Cayley did it in 1879. What his solution also contains is a line on which the method never settles — and a change of variable shows that on that line Newton's method is exactly the doubling map, the simplest chaos there is. Run on the real line for x² + 1, which has no real root to find, it becomes a source of Cauchy-distributed randomness.

dynamics · Newton basins

Named alongside it

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

Periodic orbitBinary expansionConjugacyDense orbitOrbitSensitive dependenceBasin of attractionBinaryCausticChaosEnvelopeEpicycloid

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