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Concept

Rotation number

The average amount by which a map advances a point around a circle, per step. It exists for every such map, never falls as the parameter rises, and sticks at each rational over a whole interval because a periodic orbit there attracts.

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

The rotation number against the parameter at K = 1. The measured rotation number of a circle map plotted against its parameter, forming a staircase that is flat over an interval at each simple rational.

The staircase that is flat almost everywhere

A map of the circle advances by an average amount each step. Plot that average against the parameter driving it and the graph is flat over an interval at every rational, rises only on a set of measure zero, and still climbs from nothing to one.

dynamics · Mode locking Two periodic orbits meeting at the edge of the 1/2 plateau. The circle at 4 parameter values, with the period-2 points marked — two orbits inside the plateau, drawing closer together towards its edge, and none outside it.

How a lock comes apart

Inside a plateau the map has two periodic orbits, one attracting and one pushing away. Track them to the plateau's edge and they run into each other and vanish together — so the edge of a tongue is a collision, and the width of the plateau is how far the two can be pulled apart.

dynamics · Mode locking The coordinate change that makes an unlocked map a rotation. The conjugating map built from one orbit, plotted as a staircase from the circle to itself, with the rigid rotation it turns the circle map into.

A rotation in different coordinates

At an unlocked parameter the map is not merely like a rigid rotation; it is one, after a change of coordinates built out of a single orbit. The theorem needs the map to be smooth enough, and the map that shows why is one whose orbit leaves a hole.

dynamics · Mode locking Two staircases and the interval between them, at K = 1.5. A plot against the drive of the upper and lower ends of the circle map's rotation interval above the critical line, two stepped curves with the band between them shaded.

A whole interval of speeds

Below the critical line every orbit of the circle map goes round at the same average speed. Above it the map folds back on itself, and the speed depends on where the orbit starts — not a few different values but a whole interval of them, every fraction in it the speed of some periodic orbit, and almost none of them ever seen by an orbit started at random.

dynamics · Mode locking The standard map as its kick grows. Three square phase portraits of the standard map at increasing strengths, dotted with orbits: curves spanning the square at the smallest, fewer at the critical value, and a scattered sea with islands at the largest.

The last circle to break

Kick a spinning rotor once a turn and most of its motions stay on curves that wind round forever, walls no orbit can cross. As the kick grows the walls break one by one, and the last to go is the one whose winding is the golden ratio — at a kick of 0.9716, found by watching a sequence of periodic orbits approximate it and asking whether they are stable.

dynamics · Mode locking

Named alongside it

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

OrbitCircle mapPeriodic orbitStabilityBifurcationSelf-similarityChaosIrrational rotationMeasureParameterAttractorConjugacy

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