Matpic
mathematics, in pictures
Concept

Circle map

A rule advancing a point around a circle by an amount depending on where it is. The average advance per step is its rotation number, and for a range of parameters that average sticks at a rational value because a periodic orbit attracts.

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

Also named here as rotation number — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

OrbitRotation numberSelf-similarityBifurcationIrrational rotationMeasureParameterPeriodic orbitStabilityConjugacyCounterexample

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