Concept

Parameter

A quantity held fixed while a system runs and varied between runs, so that a family of behaviours is indexed by it. Plotting a measured quantity against one is how a bifurcation, a plateau or a threshold becomes visible at all.

Named by 2 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

Named alongside it

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

Bifurcation Circle map Orbit Periodic orbit Rotation number Self-similarity Stability Irrational rotation Measure

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