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Mode locking — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · dynamics
  2. 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.

    part 2 · dynamics
  3. 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.

    part 3 · dynamics
  4. 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.

    part 4 · dynamics
  5. 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.

    part 5 · dynamics
  6. The coordinate change that makes the map a rotation. Graphs of the departure from the identity of the conjugacy between the circle map and the golden rotation, at three nonlinearities, each smooth and growing steeper with K.

    How smooth the disguise is

    An unlocked circle map is a rigid rotation in different coordinates, and whether those coordinates are smooth is decided by arithmetic. Solving for the change of coordinates means dividing, harmonic by harmonic, by numbers that come arbitrarily close to nought — and the golden rotation number keeps them far enough away that the result is not merely smooth but analytic, while a number lying close to a fraction makes it buckle.

    part 6 · dynamics

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